/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 245 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 59.7 s] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 25 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 15 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 89 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 28 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 16 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 10 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 26 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), 7 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 29 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 18 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 31 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(minus, x), 0) -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0) -> app(s, 0) app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0) -> 0 app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(minus, x), 0) -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0) -> app(s, 0) app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0) -> 0 app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(minus, x), 0) -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0) -> app(s, 0) app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0) -> 0 app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(minus, x), 0) -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0) -> app(s, 0) app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0) -> 0 app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(minus, x), 0') -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0') -> app(s, 0') app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0') -> 0' app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0') -> 0' encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0' encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(app(minus, x), 0') -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0') -> app(s, 0') app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0') -> 0' app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0') -> 0' encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0' encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app minus :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app 0' :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app s :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app f :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app g :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app map :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app nil :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app cons :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app filter :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app true :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app false :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encArg :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_minus :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_0 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_s :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app hole_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: app(app(minus, x), 0') -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0') -> app(s, 0') app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0') -> 0' app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0') -> 0' encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0' encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app minus :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app 0' :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app s :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app f :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app g :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app map :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app nil :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app cons :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app filter :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app true :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app false :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encArg :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_minus :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_0 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_s :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_g :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app hole_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(0) <=> minus gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(x, 1)) <=> cons_app(minus, gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(x)) The following defined symbols remain to be analysed: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(n58755_3)) -> *3_3, rt in Omega(0) Induction Base: encArg(gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(0)) Induction Step: encArg(gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(n58755_3, 1))) ->_R^Omega(0) app(encArg(minus), encArg(gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(n58755_3))) ->_R^Omega(0) app(minus, encArg(gen_minus:0':s:f:g:map:nil:cons:filter:filter2:true:false:cons_app2_3(n58755_3))) ->_IH app(minus, *3_3) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF) ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(minus, x), 0) -> x app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y) app(f, 0) -> app(s, 0) app(f, app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(g, 0) -> 0 app(g, app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, 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(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(f, 0) -> app(s, 0) app(g, 0) -> 0 app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(f, c_app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) app(c_app(map, fun), nil) -> nil app(c_app(filter, fun), nil) -> nil app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(c_app(minus, x), 0) -> x app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(app(minus, x), y) app(g, c_app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x))) The (relative) TRS S consists of the following rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(x0, x1) -> c_app(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(f, 0) -> app(s, 0) [1] app(g, 0) -> 0 [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(f, c_app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(app(minus, x), y) [1] app(g, c_app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x))) [1] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(s) -> s [0] encArg(f) -> f [0] encArg(g) -> g [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_s -> s [0] encode_f -> f [0] encode_g -> g [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(f, 0) -> app(s, 0) [1] app(g, 0) -> 0 [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(f, c_app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(app(minus, x), y) [1] app(g, c_app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x))) [1] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(s) -> s [0] encArg(f) -> f [0] encArg(g) -> g [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_s -> s [0] encode_f -> f [0] encode_g -> g [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] The TRS has the following type information: app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app 0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app c_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app cons_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app Rewrite Strategy: INNERMOST ---------------------------------------- (23) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_app_2 encode_minus encode_0 encode_s encode_f encode_g encode_map encode_nil encode_cons encode_filter encode_filter2 encode_true encode_false app_2 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s -> null_encode_s [0] encode_f -> null_encode_f [0] encode_g -> null_encode_g [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_minus, null_encode_0, null_encode_s, null_encode_f, null_encode_g, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(f, 0) -> app(s, 0) [1] app(g, 0) -> 0 [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(f, c_app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(app(minus, x), y) [1] app(g, c_app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x))) [1] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(s) -> s [0] encArg(f) -> f [0] encArg(g) -> g [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_s -> s [0] encode_f -> f [0] encode_g -> g [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s -> null_encode_s [0] encode_f -> null_encode_f [0] encode_g -> null_encode_g [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(f, 0) -> app(s, 0) [1] app(g, 0) -> 0 [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(f, c_app(s, 0)) -> app(app(minus, c_app(s, 0)), app(g, app(s, 0))) [2] app(f, c_app(s, c_app(s, x8))) -> app(app(minus, c_app(s, c_app(s, x8))), app(g, app(app(minus, app(s, x8)), app(g, app(f, x8))))) [2] app(f, c_app(s, x)) -> app(app(minus, c_app(s, x)), app(g, c_app(f, x))) [1] app(f, c_app(s, x)) -> app(app(minus, c_app(s, x)), app(g, null_app)) [1] app(f, c_app(s, 0)) -> app(app(minus, null_app), app(g, app(s, 0))) [2] app(f, c_app(s, c_app(s, x9))) -> app(app(minus, null_app), app(g, app(app(minus, app(s, x9)), app(g, app(f, x9))))) [2] app(f, c_app(s, x)) -> app(app(minus, null_app), app(g, c_app(f, x))) [1] app(f, c_app(s, x)) -> app(app(minus, null_app), app(g, null_app)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(c_app(cons, x), app(c_app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(c_app(cons, x), app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(null_app, app(c_app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(null_app, app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(c_app(filter, fun), xs) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(null_app, xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, f), c_app(c_app(cons, 0), xs)) -> app(app(app(app(filter2, app(s, 0)), f), 0), xs) [2] app(c_app(filter, g), c_app(c_app(cons, 0), xs)) -> app(app(app(app(filter2, 0), g), 0), xs) [2] app(c_app(filter, c_app(map, fun5)), c_app(c_app(cons, c_app(c_app(cons, x10), xs1)), xs)) -> app(app(app(app(filter2, app(app(cons, app(fun5, x10)), app(app(map, fun5), xs1))), c_app(map, fun5)), c_app(c_app(cons, x10), xs1)), xs) [2] app(c_app(filter, f), c_app(c_app(cons, c_app(s, x11)), xs)) -> app(app(app(app(filter2, app(app(minus, app(s, x11)), app(g, app(f, x11)))), f), c_app(s, x11)), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, true), fun6), x12)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(cons, x12), app(app(filter, fun6), x))), c_app(c_app(c_app(filter2, true), fun6), x12)), x), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, false), fun7), x13)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(filter, fun7), x)), c_app(c_app(c_app(filter2, false), fun7), x13)), x), xs) [2] app(c_app(filter, c_app(map, fun8)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(map, fun8)), nil), xs) [2] app(c_app(filter, c_app(filter, fun9)), c_app(c_app(cons, nil), xs)) -> app(app(app(app(filter2, nil), c_app(filter, fun9)), nil), xs) [2] app(c_app(filter, c_app(filter, fun10)), c_app(c_app(cons, c_app(c_app(cons, x14), xs2)), xs)) -> app(app(app(app(filter2, app(app(app(app(filter2, app(fun10, x14)), fun10), x14), xs2)), c_app(filter, fun10)), c_app(c_app(cons, x14), xs2)), xs) [2] app(c_app(filter, c_app(minus, x15)), c_app(c_app(cons, 0), xs)) -> app(app(app(app(filter2, x15), c_app(minus, x15)), 0), xs) [2] app(c_app(filter, c_app(minus, c_app(s, x16))), c_app(c_app(cons, c_app(s, y'')), xs)) -> app(app(app(app(filter2, app(app(minus, x16), y'')), c_app(minus, c_app(s, x16))), c_app(s, y'')), xs) [2] app(c_app(filter, g), c_app(c_app(cons, c_app(s, x17)), xs)) -> app(app(app(app(filter2, app(app(minus, app(s, x17)), app(f, app(g, x17)))), g), c_app(s, x17)), xs) [2] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, c_app(fun, x)), fun), x), xs) [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, null_app), fun), x), xs) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(c_app(minus, x), y) [1] app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(null_app, y) [1] app(g, c_app(s, 0)) -> app(app(minus, c_app(s, 0)), app(f, 0)) [2] app(g, c_app(s, c_app(s, x18))) -> app(app(minus, c_app(s, c_app(s, x18))), app(f, app(app(minus, app(s, x18)), app(f, app(g, x18))))) [2] app(g, c_app(s, x)) -> app(app(minus, c_app(s, x)), app(f, c_app(g, x))) [1] app(g, c_app(s, x)) -> app(app(minus, c_app(s, x)), app(f, null_app)) [1] app(g, c_app(s, 0)) -> app(app(minus, null_app), app(f, 0)) [2] app(g, c_app(s, c_app(s, x19))) -> app(app(minus, null_app), app(f, app(app(minus, app(s, x19)), app(f, app(g, x19))))) [2] app(g, c_app(s, x)) -> app(app(minus, null_app), app(f, c_app(g, x))) [1] app(g, c_app(s, x)) -> app(app(minus, null_app), app(f, null_app)) [1] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(s) -> s [0] encArg(f) -> f [0] encArg(g) -> g [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_s -> s [0] encode_f -> f [0] encode_g -> g [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s -> null_encode_s [0] encode_f -> null_encode_f [0] encode_g -> null_encode_g [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: f => 2 0 => 0 s => 10 g => 6 map => 7 cons => 1 minus => 8 filter2 => 5 true => 11 filter => 4 false => 3 nil => 9 null_encArg => 0 null_encode_app => 0 null_encode_minus => 0 null_encode_0 => 0 null_encode_s => 0 null_encode_f => 0 null_encode_g => 0 null_encode_map => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filter => 0 null_encode_filter2 => 0 null_encode_true => 0 null_encode_false => 0 null_app => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> x :|: x >= 0, z = 1 + 8 + x, z' = 0 app(z, z') -{ 2 }-> app(app(app(app(5, x15), 1 + 8 + x15), 0), xs) :|: xs >= 0, z = 1 + 4 + (1 + 8 + x15), z' = 1 + (1 + 1 + 0) + xs, x15 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(app(app(5, app(fun10, x14)), fun10), x14), xs2)), 1 + 4 + fun10), 1 + (1 + 1 + x14) + xs2), xs) :|: xs >= 0, xs2 >= 0, z = 1 + 4 + (1 + 4 + fun10), z' = 1 + (1 + 1 + (1 + (1 + 1 + x14) + xs2)) + xs, x14 >= 0, fun10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(8, x16), y'')), 1 + 8 + (1 + 10 + x16)), 1 + 10 + y''), xs) :|: xs >= 0, z = 1 + 4 + (1 + 8 + (1 + 10 + x16)), y'' >= 0, z' = 1 + (1 + 1 + (1 + 10 + y'')) + xs, x16 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(8, app(10, x11)), app(6, app(2, x11)))), 2), 1 + 10 + x11), xs) :|: xs >= 0, z' = 1 + (1 + 1 + (1 + 10 + x11)) + xs, z = 1 + 4 + 2, x11 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(8, app(10, x17)), app(2, app(6, x17)))), 6), 1 + 10 + x17), xs) :|: xs >= 0, x17 >= 0, z = 1 + 4 + 6, z' = 1 + (1 + 1 + (1 + 10 + x17)) + xs app(z, z') -{ 2 }-> app(app(app(app(5, app(app(4, fun7), x)), 1 + (1 + (1 + 5 + 3) + fun7) + x13), x), xs) :|: x13 >= 0, xs >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 3) + fun7) + x13), x >= 0, z' = 1 + (1 + 1 + x) + xs, fun7 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, x12), app(app(4, fun6), x))), 1 + (1 + (1 + 5 + 11) + fun6) + x12), x), xs) :|: fun6 >= 0, xs >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 11) + fun6) + x12), x >= 0, x12 >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, app(fun5, x10)), app(app(7, fun5), xs1))), 1 + 7 + fun5), 1 + (1 + 1 + x10) + xs1), xs) :|: xs >= 0, z' = 1 + (1 + 1 + (1 + (1 + 1 + x10) + xs1)) + xs, xs1 >= 0, z = 1 + 4 + (1 + 7 + fun5), fun5 >= 0, x10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(10, 0)), 2), 0), xs) :|: xs >= 0, z = 1 + 4 + 2, z' = 1 + (1 + 1 + 0) + xs app(z, z') -{ 2 }-> app(app(app(app(5, 9), 1 + 7 + fun8), 9), xs) :|: z' = 1 + (1 + 1 + 9) + xs, xs >= 0, fun8 >= 0, z = 1 + 4 + (1 + 7 + fun8) app(z, z') -{ 2 }-> app(app(app(app(5, 9), 1 + 4 + fun9), 9), xs) :|: z' = 1 + (1 + 1 + 9) + xs, xs >= 0, fun9 >= 0, z = 1 + 4 + (1 + 4 + fun9) app(z, z') -{ 1 }-> app(app(app(app(5, 0), fun), x), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 4 + fun, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(5, 0), 6), 0), xs) :|: xs >= 0, z = 1 + 4 + 6, z' = 1 + (1 + 1 + 0) + xs app(z, z') -{ 1 }-> app(app(app(app(5, 1 + fun + x), fun), x), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 4 + fun, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(8, 0), app(6, app(app(8, app(10, x9)), app(6, app(2, x9))))) :|: z = 2, z' = 1 + 10 + (1 + 10 + x9), x9 >= 0 app(z, z') -{ 2 }-> app(app(8, 0), app(6, app(10, 0))) :|: z = 2, z' = 1 + 10 + 0 app(z, z') -{ 1 }-> app(app(8, 0), app(6, 0)) :|: z = 2, x >= 0, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, 0), app(6, 1 + 2 + x)) :|: z = 2, x >= 0, z' = 1 + 10 + x app(z, z') -{ 2 }-> app(app(8, 0), app(2, app(app(8, app(10, x19)), app(2, app(6, x19))))) :|: z' = 1 + 10 + (1 + 10 + x19), x19 >= 0, z = 6 app(z, z') -{ 2 }-> app(app(8, 0), app(2, 0)) :|: z' = 1 + 10 + 0, z = 6 app(z, z') -{ 1 }-> app(app(8, 0), app(2, 0)) :|: x >= 0, z = 6, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, 0), app(2, 1 + 6 + x)) :|: x >= 0, z = 6, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, 1 + 10 + x), app(6, 0)) :|: z = 2, x >= 0, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, 1 + 10 + x), app(6, 1 + 2 + x)) :|: z = 2, x >= 0, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, 1 + 10 + x), app(2, 0)) :|: x >= 0, z = 6, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, 1 + 10 + x), app(2, 1 + 6 + x)) :|: x >= 0, z = 6, z' = 1 + 10 + x app(z, z') -{ 2 }-> app(app(8, 1 + 10 + 0), app(6, app(10, 0))) :|: z = 2, z' = 1 + 10 + 0 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + 0), app(2, 0)) :|: z' = 1 + 10 + 0, z = 6 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + (1 + 10 + x18)), app(2, app(app(8, app(10, x18)), app(2, app(6, x18))))) :|: z' = 1 + 10 + (1 + 10 + x18), z = 6, x18 >= 0 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + (1 + 10 + x8)), app(6, app(app(8, app(10, x8)), app(6, app(2, x8))))) :|: z = 2, z' = 1 + 10 + (1 + 10 + x8), x8 >= 0 app(z, z') -{ 1 }-> app(app(1, app(fun, x)), app(app(7, fun), xs)) :|: fun >= 0, xs >= 0, z = 1 + 7 + fun, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(10, 0) :|: z = 2, z' = 0 app(z, z') -{ 1 }-> app(0, xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 3) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(0, y) :|: z = 1 + 8 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> app(0, app(0, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(0, app(1 + 4 + fun, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(1 + 8 + x, y) :|: z = 1 + 8 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> app(1 + 4 + fun, xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 3) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(1 + 1 + x, app(0, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(1 + 1 + x, app(1 + 4 + fun, xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x, z' = xs app(z, z') -{ 1 }-> 9 :|: fun >= 0, z = 1 + 7 + fun, z' = 9 app(z, z') -{ 1 }-> 9 :|: fun >= 0, z = 1 + 4 + fun, z' = 9 app(z, z') -{ 1 }-> 0 :|: z = 6, z' = 0 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 9 :|: z = 9 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 11 :|: z = 11 encArg(z) -{ 0 }-> 10 :|: z = 10 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 2 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 3 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 4 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 5 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 6 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 7 :|: encode_map -{ 0 }-> 0 :|: encode_minus -{ 0 }-> 8 :|: encode_minus -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 9 :|: encode_nil -{ 0 }-> 0 :|: encode_s -{ 0 }-> 10 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 11 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 2 }-> app(app(app(app(5, app(app(app(app(5, app(z - 10, x14)), z - 10), x14), xs2)), 1 + 4 + (z - 10)), 1 + (1 + 1 + x14) + xs2), xs) :|: xs >= 0, xs2 >= 0, z' = 1 + (1 + 1 + (1 + (1 + 1 + x14) + xs2)) + xs, x14 >= 0, z - 10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(8, app(10, x11)), app(6, app(2, x11)))), 2), 1 + 10 + x11), xs) :|: xs >= 0, z' = 1 + (1 + 1 + (1 + 10 + x11)) + xs, z = 1 + 4 + 2, x11 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(8, app(10, x17)), app(2, app(6, x17)))), 6), 1 + 10 + x17), xs) :|: xs >= 0, x17 >= 0, z = 1 + 4 + 6, z' = 1 + (1 + 1 + (1 + 10 + x17)) + xs app(z, z') -{ 2 }-> app(app(app(app(5, app(app(8, z - 25), y'')), 1 + 8 + (1 + 10 + (z - 25))), 1 + 10 + y''), xs) :|: xs >= 0, y'' >= 0, z' = 1 + (1 + 1 + (1 + 10 + y'')) + xs, z - 25 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(4, fun7), x)), 1 + (1 + (1 + 5 + 3) + fun7) + x13), x), xs) :|: x13 >= 0, xs >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 3) + fun7) + x13), x >= 0, z' = 1 + (1 + 1 + x) + xs, fun7 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, x12), app(app(4, fun6), x))), 1 + (1 + (1 + 5 + 11) + fun6) + x12), x), xs) :|: fun6 >= 0, xs >= 0, z = 1 + 4 + (1 + (1 + (1 + 5 + 11) + fun6) + x12), x >= 0, x12 >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(5, app(app(1, app(z - 13, x10)), app(app(7, z - 13), xs1))), 1 + 7 + (z - 13)), 1 + (1 + 1 + x10) + xs1), xs) :|: xs >= 0, z' = 1 + (1 + 1 + (1 + (1 + 1 + x10) + xs1)) + xs, xs1 >= 0, z - 13 >= 0, x10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, app(10, 0)), 2), 0), z' - 3) :|: z' - 3 >= 0, z = 1 + 4 + 2 app(z, z') -{ 2 }-> app(app(app(app(5, 9), 1 + 7 + (z - 13)), 9), z' - 12) :|: z' - 12 >= 0, z - 13 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, 9), 1 + 4 + (z - 10)), 9), z' - 12) :|: z' - 12 >= 0, z - 10 >= 0 app(z, z') -{ 2 }-> app(app(app(app(5, 0), 6), 0), z' - 3) :|: z' - 3 >= 0, z = 1 + 4 + 6 app(z, z') -{ 1 }-> app(app(app(app(5, 0), z - 5), x), xs) :|: z - 5 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(app(app(5, z - 14), 1 + 8 + (z - 14)), 0), z' - 3) :|: z' - 3 >= 0, z - 14 >= 0 app(z, z') -{ 1 }-> app(app(app(app(5, 1 + (z - 5) + x), z - 5), x), xs) :|: z - 5 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 2 }-> app(app(8, 0), app(6, app(app(8, app(10, z' - 22)), app(6, app(2, z' - 22))))) :|: z = 2, z' - 22 >= 0 app(z, z') -{ 2 }-> app(app(8, 0), app(6, app(10, 0))) :|: z = 2, z' = 1 + 10 + 0 app(z, z') -{ 1 }-> app(app(8, 0), app(6, 0)) :|: z = 2, z' - 11 >= 0 app(z, z') -{ 1 }-> app(app(8, 0), app(6, 1 + 2 + (z' - 11))) :|: z = 2, z' - 11 >= 0 app(z, z') -{ 2 }-> app(app(8, 0), app(2, app(app(8, app(10, z' - 22)), app(2, app(6, z' - 22))))) :|: z' - 22 >= 0, z = 6 app(z, z') -{ 2 }-> app(app(8, 0), app(2, 0)) :|: z' = 1 + 10 + 0, z = 6 app(z, z') -{ 1 }-> app(app(8, 0), app(2, 0)) :|: z' - 11 >= 0, z = 6 app(z, z') -{ 1 }-> app(app(8, 0), app(2, 1 + 6 + (z' - 11))) :|: z' - 11 >= 0, z = 6 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + 0), app(6, app(10, 0))) :|: z = 2, z' = 1 + 10 + 0 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + 0), app(2, 0)) :|: z' = 1 + 10 + 0, z = 6 app(z, z') -{ 1 }-> app(app(8, 1 + 10 + (z' - 11)), app(6, 0)) :|: z = 2, z' - 11 >= 0 app(z, z') -{ 1 }-> app(app(8, 1 + 10 + (z' - 11)), app(6, 1 + 2 + (z' - 11))) :|: z = 2, z' - 11 >= 0 app(z, z') -{ 1 }-> app(app(8, 1 + 10 + (z' - 11)), app(2, 0)) :|: z' - 11 >= 0, z = 6 app(z, z') -{ 1 }-> app(app(8, 1 + 10 + (z' - 11)), app(2, 1 + 6 + (z' - 11))) :|: z' - 11 >= 0, z = 6 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + (1 + 10 + (z' - 22))), app(6, app(app(8, app(10, z' - 22)), app(6, app(2, z' - 22))))) :|: z = 2, z' - 22 >= 0 app(z, z') -{ 2 }-> app(app(8, 1 + 10 + (1 + 10 + (z' - 22))), app(2, app(app(8, app(10, z' - 22)), app(2, app(6, z' - 22))))) :|: z = 6, z' - 22 >= 0 app(z, z') -{ 1 }-> app(app(1, app(z - 8, x)), app(app(7, z - 8), xs)) :|: z - 8 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(10, 0) :|: z = 2, z' = 0 app(z, z') -{ 1 }-> app(0, z') :|: fun >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 3) + fun) + x app(z, z') -{ 1 }-> app(0, app(0, z')) :|: fun >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x app(z, z') -{ 1 }-> app(0, app(1 + 4 + fun, z')) :|: fun >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x app(z, z') -{ 1 }-> app(0, z' - 11) :|: z - 20 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(1 + 8 + (z - 20), z' - 11) :|: z - 20 >= 0, z' - 11 >= 0 app(z, z') -{ 1 }-> app(1 + 4 + fun, z') :|: fun >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 3) + fun) + x app(z, z') -{ 1 }-> app(1 + 1 + x, app(0, z')) :|: fun >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x app(z, z') -{ 1 }-> app(1 + 1 + x, app(1 + 4 + fun, z')) :|: fun >= 0, z' >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x app(z, z') -{ 1 }-> 9 :|: z - 8 >= 0, z' = 9 app(z, z') -{ 1 }-> 9 :|: z - 5 >= 0, z' = 9 app(z, z') -{ 1 }-> 0 :|: z = 6, z' = 0 app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 app(z, z') -{ 1 }-> z - 9 :|: z - 9 >= 0, z' = 0 app(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 9 :|: z = 9 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 11 :|: z = 11 encArg(z) -{ 0 }-> 10 :|: z = 10 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 2 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 3 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 4 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 5 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 6 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 7 :|: encode_map -{ 0 }-> 0 :|: encode_minus -{ 0 }-> 8 :|: encode_minus -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 9 :|: encode_nil -{ 0 }-> 0 :|: encode_s -{ 0 }-> 10 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 11 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s -> null_encode_s [0] encode_f -> null_encode_f [0] encode_g -> null_encode_g [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_minus, null_encode_0, null_encode_s, null_encode_f, null_encode_g, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(f, 0) -> app(s, 0) [1] app(g, 0) -> 0 [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(f, c_app(s, x)) -> app(app(minus, app(s, x)), app(g, app(f, x))) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(minus, x), 0) -> x [1] app(c_app(minus, c_app(s, x)), c_app(s, y)) -> app(app(minus, x), y) [1] app(g, c_app(s, x)) -> app(app(minus, app(s, x)), app(f, app(g, x))) [1] encArg(minus) -> minus [0] encArg(0) -> 0 [0] encArg(s) -> s [0] encArg(f) -> f [0] encArg(g) -> g [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_minus -> minus [0] encode_0 -> 0 [0] encode_s -> s [0] encode_f -> f [0] encode_g -> g [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_minus -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s -> null_encode_s [0] encode_f -> null_encode_f [0] encode_g -> null_encode_g [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_minus :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_0 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_s :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_f :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_g :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: f:0:s:g:map:c_app:cons:minus:filter2:true:filter:false:nil:cons_app:null_encArg:null_encode_app:null_encode_minus:null_encode_0:null_encode_s:null_encode_f:null_encode_g:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: f => 2 0 => 0 s => 10 g => 6 map => 7 cons => 1 minus => 8 filter2 => 5 true => 11 filter => 4 false => 3 nil => 9 null_encArg => 0 null_encode_app => 0 null_encode_minus => 0 null_encode_0 => 0 null_encode_s => 0 null_encode_f => 0 null_encode_g => 0 null_encode_map => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filter => 0 null_encode_filter2 => 0 null_encode_true => 0 null_encode_false => 0 null_app => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> x :|: x >= 0, z = 1 + 8 + x, z' = 0 app(z, z') -{ 1 }-> app(app(app(app(5, app(fun, x)), fun), x), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + 4 + fun, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(8, x), y) :|: z = 1 + 8 + (1 + 10 + x), x >= 0, y >= 0, z' = 1 + 10 + y app(z, z') -{ 1 }-> app(app(8, app(10, x)), app(6, app(2, x))) :|: z = 2, x >= 0, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(8, app(10, x)), app(2, app(6, x))) :|: x >= 0, z = 6, z' = 1 + 10 + x app(z, z') -{ 1 }-> app(app(4, fun), xs) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 3) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(app(1, x), app(app(4, fun), xs)) :|: fun >= 0, xs >= 0, x >= 0, z = 1 + (1 + (1 + 5 + 11) + fun) + x, z' = xs app(z, z') -{ 1 }-> app(app(1, app(fun, x)), app(app(7, fun), xs)) :|: fun >= 0, xs >= 0, z = 1 + 7 + fun, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(10, 0) :|: z = 2, z' = 0 app(z, z') -{ 1 }-> 9 :|: fun >= 0, z = 1 + 7 + fun, z' = 9 app(z, z') -{ 1 }-> 9 :|: fun >= 0, z = 1 + 4 + fun, z' = 9 app(z, z') -{ 1 }-> 0 :|: z = 6, z' = 0 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 9 :|: z = 9 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 11 :|: z = 11 encArg(z) -{ 0 }-> 10 :|: z = 10 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 2 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 3 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 4 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 5 :|: encode_filter2 -{ 0 }-> 0 :|: encode_g -{ 0 }-> 6 :|: encode_g -{ 0 }-> 0 :|: encode_map -{ 0 }-> 7 :|: encode_map -{ 0 }-> 0 :|: encode_minus -{ 0 }-> 8 :|: encode_minus -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 9 :|: encode_nil -{ 0 }-> 0 :|: encode_s -{ 0 }-> 10 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 11 :|: encode_true -{ 0 }-> 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(minus) -> c ENCARG(0) -> c1 ENCARG(s) -> c2 ENCARG(f) -> c3 ENCARG(g) -> c4 ENCARG(map) -> c5 ENCARG(nil) -> c6 ENCARG(cons) -> c7 ENCARG(filter) -> c8 ENCARG(filter2) -> c9 ENCARG(true) -> c10 ENCARG(false) -> c11 ENCARG(cons_app(z0, z1)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c13(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_MINUS -> c14 ENCODE_0 -> c15 ENCODE_S -> c16 ENCODE_F -> c17 ENCODE_G -> c18 ENCODE_MAP -> c19 ENCODE_NIL -> c20 ENCODE_CONS -> c21 ENCODE_FILTER -> c22 ENCODE_FILTER2 -> c23 ENCODE_TRUE -> c24 ENCODE_FALSE -> c25 APP(app(minus, z0), 0) -> c26 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1), APP(minus, z0)) APP(f, 0) -> c28(APP(s, 0)) APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(g, app(f, z0)), APP(f, z0)) APP(g, 0) -> c30 APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), nil) -> c32 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(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) -> c34 APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(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) -> c36(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) -> c37(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1), APP(minus, z0)) APP(f, 0) -> c28(APP(s, 0)) APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(g, app(f, z0)), APP(f, z0)) APP(g, 0) -> c30 APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), nil) -> c32 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(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) -> c34 APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(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) -> c36(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) -> c37(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_minus, encode_0, encode_s, encode_f, encode_g, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_MINUS, ENCODE_0, ENCODE_S, ENCODE_F, ENCODE_G, ENCODE_MAP, ENCODE_NIL, ENCODE_CONS, ENCODE_FILTER, ENCODE_FILTER2, ENCODE_TRUE, ENCODE_FALSE, APP_2 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12_3, c13_3, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27_2, c28_1, c29_5, c30, c31_5, c32, c33_5, c34, c35_5, c36_4, c37_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 24 trailing nodes: ENCARG(s) -> c2 ENCARG(false) -> c11 ENCODE_NIL -> c20 ENCARG(true) -> c10 ENCODE_CONS -> c21 ENCODE_F -> c17 ENCODE_S -> c16 ENCODE_FILTER2 -> c23 ENCARG(filter) -> c8 ENCARG(cons) -> c7 ENCODE_0 -> c15 ENCARG(map) -> c5 ENCODE_FALSE -> c25 ENCODE_MINUS -> c14 ENCARG(minus) -> c ENCODE_FILTER -> c22 ENCODE_TRUE -> c24 ENCODE_MAP -> c19 ENCARG(g) -> c4 ENCARG(f) -> c3 ENCARG(0) -> c1 ENCARG(nil) -> c6 ENCODE_G -> c18 ENCARG(filter2) -> c9 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c13(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(minus, z0), 0) -> c26 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1), APP(minus, z0)) APP(f, 0) -> c28(APP(s, 0)) APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(g, app(f, z0)), APP(f, z0)) APP(g, 0) -> c30 APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), nil) -> c32 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(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) -> c34 APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(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) -> c36(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) -> c37(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1), APP(minus, z0)) APP(f, 0) -> c28(APP(s, 0)) APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(g, app(f, z0)), APP(f, z0)) APP(g, 0) -> c30 APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(minus, app(s, z0)), APP(s, z0), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), nil) -> c32 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(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) -> c34 APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(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) -> c36(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) -> c37(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_minus, encode_0, encode_s, encode_f, encode_g, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c12_3, c13_3, c26, c27_2, c28_1, c29_5, c30, c31_5, c32, c33_5, c34, c35_5, c36_4, c37_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c13(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_minus, encode_0, encode_s, encode_f, encode_g, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c12_3, c13_3, c26, c30, c32, c34, c27_1, c28, c29_3, c31_3, c33_2, c35_2, c36_1, c37_1 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_minus, encode_0, encode_s, encode_f, encode_g, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c12_3, c26, c30, c32, c34, c27_1, c28, c29_3, c31_3, c33_2, c35_2, c36_1, c37_1, c_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_minus, encode_0, encode_s, encode_f, encode_g, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c12_3, c26, c30, c32, c34, c27_1, c28, c29_3, c31_3, c33_2, c35_2, c36_1, c37_1, c_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_minus -> minus encode_0 -> 0 encode_s -> s encode_f -> f encode_g -> g encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c12_3, c26, c30, c32, c34, c27_1, c28, c29_3, c31_3, c33_2, c35_2, c36_1, c37_1, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0), ENCARG(minus)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(minus, x1)) -> c12(APP(minus, encArg(x1)), ENCARG(minus), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0), ENCARG(minus)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0), ENCARG(g)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(minus, x1)) -> c12(APP(minus, encArg(x1)), ENCARG(minus), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(g), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c29_3, c31_3, c33_2, c35_2, c36_1, c37_1, c_1, c12_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 34 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c29_3, c31_3, c33_2, c35_2, c36_1, c37_1, c_1, c12_3, c12_2, c12_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(f, app(s, z0)) -> c29(APP(app(minus, app(s, z0)), app(g, app(f, z0))), APP(g, app(f, z0)), APP(f, z0)) by APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c31_3, c33_2, c35_2, c36_1, c37_1, c_1, c12_3, c12_2, c12_1, c29_3 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(g, app(s, z0)) -> c31(APP(app(minus, app(s, z0)), app(f, app(g, z0))), APP(f, app(g, z0)), APP(g, z0)) by APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c35_2, c36_1, c37_1, c_1, c12_3, c12_2, c12_1, c29_3, c31_3 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c35(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, app(s, 0)), f), 0), x2), APP(f, 0)) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, 0), g), 0), x2), APP(g, 0)) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(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)) -> c35(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)) -> c35(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)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c35(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)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, app(s, 0)), f), 0), x2), APP(f, 0)) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, 0), g), 0), x2), APP(g, 0)) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(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)) -> c35(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)) -> c35(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)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c35(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)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, app(s, 0)), f), 0), x2), APP(f, 0)) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, 0), g), 0), x2), APP(g, 0)) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(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)) -> c35(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)) -> c35(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)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c35(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)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c_1, c12_3, c12_2, c12_1, c29_3, c31_3, c35_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c_1, c12_3, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(minus, x1) -> c(APP(minus, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(minus, x1) -> c(APP(minus, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_3, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(minus, x1) -> c(APP(minus, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_3, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c12(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(APP(minus, app(encArg(x1), encArg(x2))), ENCARG(minus), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(APP(minus, app(encArg(x1), encArg(x2))), ENCARG(minus), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(g), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_3, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 22 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_3, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c12(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1)), ENCARG(minus)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c12(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1))), ENCARG(cons_app(x0, x1)), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1)), ENCARG(minus)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1)), ENCARG(g)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 19 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, minus)) -> c12(APP(encArg(x0), minus), ENCARG(x0)) by ENCARG(cons_app(minus, minus)) -> c12(APP(minus, minus), ENCARG(minus)) ENCARG(cons_app(0, minus)) -> c12(APP(0, minus), ENCARG(0)) ENCARG(cons_app(s, minus)) -> c12(APP(s, minus), ENCARG(s)) ENCARG(cons_app(f, minus)) -> c12(APP(f, minus), ENCARG(f)) ENCARG(cons_app(g, minus)) -> c12(APP(g, minus), ENCARG(g)) ENCARG(cons_app(map, minus)) -> c12(APP(map, minus), ENCARG(map)) ENCARG(cons_app(nil, minus)) -> c12(APP(nil, minus), ENCARG(nil)) ENCARG(cons_app(cons, minus)) -> c12(APP(cons, minus), ENCARG(cons)) ENCARG(cons_app(filter, minus)) -> c12(APP(filter, minus), ENCARG(filter)) ENCARG(cons_app(filter2, minus)) -> c12(APP(filter2, minus), ENCARG(filter2)) ENCARG(cons_app(true, minus)) -> c12(APP(true, minus), ENCARG(true)) ENCARG(cons_app(false, minus)) -> c12(APP(false, minus), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), minus)) -> c12(APP(app(encArg(z0), encArg(z1)), minus), ENCARG(cons_app(z0, z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(minus, minus)) -> c12(APP(minus, minus), ENCARG(minus)) ENCARG(cons_app(0, minus)) -> c12(APP(0, minus), ENCARG(0)) ENCARG(cons_app(s, minus)) -> c12(APP(s, minus), ENCARG(s)) ENCARG(cons_app(f, minus)) -> c12(APP(f, minus), ENCARG(f)) ENCARG(cons_app(g, minus)) -> c12(APP(g, minus), ENCARG(g)) ENCARG(cons_app(map, minus)) -> c12(APP(map, minus), ENCARG(map)) ENCARG(cons_app(nil, minus)) -> c12(APP(nil, minus), ENCARG(nil)) ENCARG(cons_app(cons, minus)) -> c12(APP(cons, minus), ENCARG(cons)) ENCARG(cons_app(filter, minus)) -> c12(APP(filter, minus), ENCARG(filter)) ENCARG(cons_app(filter2, minus)) -> c12(APP(filter2, minus), ENCARG(filter2)) ENCARG(cons_app(true, minus)) -> c12(APP(true, minus), ENCARG(true)) ENCARG(cons_app(false, minus)) -> c12(APP(false, minus), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(nil, minus)) -> c12(APP(nil, minus), ENCARG(nil)) ENCARG(cons_app(filter, minus)) -> c12(APP(filter, minus), ENCARG(filter)) ENCARG(cons_app(s, minus)) -> c12(APP(s, minus), ENCARG(s)) ENCARG(cons_app(false, minus)) -> c12(APP(false, minus), ENCARG(false)) ENCARG(cons_app(map, minus)) -> c12(APP(map, minus), ENCARG(map)) ENCARG(cons_app(cons, minus)) -> c12(APP(cons, minus), ENCARG(cons)) ENCARG(cons_app(f, minus)) -> c12(APP(f, minus), ENCARG(f)) ENCARG(cons_app(minus, minus)) -> c12(APP(minus, minus), ENCARG(minus)) ENCARG(cons_app(true, minus)) -> c12(APP(true, minus), ENCARG(true)) ENCARG(cons_app(g, minus)) -> c12(APP(g, minus), ENCARG(g)) ENCARG(cons_app(0, minus)) -> c12(APP(0, minus), ENCARG(0)) ENCARG(cons_app(filter2, minus)) -> c12(APP(filter2, minus), ENCARG(filter2)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, 0)) -> c12(APP(encArg(x0), 0), ENCARG(x0)) by ENCARG(cons_app(minus, 0)) -> c12(APP(minus, 0), ENCARG(minus)) ENCARG(cons_app(0, 0)) -> c12(APP(0, 0), ENCARG(0)) ENCARG(cons_app(s, 0)) -> c12(APP(s, 0), ENCARG(s)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0), ENCARG(f)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0), ENCARG(g)) ENCARG(cons_app(map, 0)) -> c12(APP(map, 0), ENCARG(map)) ENCARG(cons_app(nil, 0)) -> c12(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c12(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filter, 0)) -> c12(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(filter2, 0)) -> c12(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(true, 0)) -> c12(APP(true, 0), ENCARG(true)) ENCARG(cons_app(false, 0)) -> c12(APP(false, 0), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), 0)) -> c12(APP(app(encArg(z0), encArg(z1)), 0), ENCARG(cons_app(z0, z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(minus, 0)) -> c12(APP(minus, 0), ENCARG(minus)) ENCARG(cons_app(0, 0)) -> c12(APP(0, 0), ENCARG(0)) ENCARG(cons_app(s, 0)) -> c12(APP(s, 0), ENCARG(s)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0), ENCARG(f)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0), ENCARG(g)) ENCARG(cons_app(map, 0)) -> c12(APP(map, 0), ENCARG(map)) ENCARG(cons_app(nil, 0)) -> c12(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c12(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filter, 0)) -> c12(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(filter2, 0)) -> c12(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(true, 0)) -> c12(APP(true, 0), ENCARG(true)) ENCARG(cons_app(false, 0)) -> c12(APP(false, 0), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCARG(cons_app(cons, 0)) -> c12(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(true, 0)) -> c12(APP(true, 0), ENCARG(true)) ENCARG(cons_app(map, 0)) -> c12(APP(map, 0), ENCARG(map)) ENCARG(cons_app(s, 0)) -> c12(APP(s, 0), ENCARG(s)) ENCARG(cons_app(0, 0)) -> c12(APP(0, 0), ENCARG(0)) ENCARG(cons_app(false, 0)) -> c12(APP(false, 0), ENCARG(false)) ENCARG(cons_app(nil, 0)) -> c12(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(filter, 0)) -> c12(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(minus, 0)) -> c12(APP(minus, 0), ENCARG(minus)) ENCARG(cons_app(filter2, 0)) -> c12(APP(filter2, 0), ENCARG(filter2)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0), ENCARG(f)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0), ENCARG(g)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, s)) -> c12(APP(encArg(x0), s), ENCARG(x0)) by ENCARG(cons_app(minus, s)) -> c12(APP(minus, s), ENCARG(minus)) ENCARG(cons_app(0, s)) -> c12(APP(0, s), ENCARG(0)) ENCARG(cons_app(s, s)) -> c12(APP(s, s), ENCARG(s)) ENCARG(cons_app(f, s)) -> c12(APP(f, s), ENCARG(f)) ENCARG(cons_app(g, s)) -> c12(APP(g, s), ENCARG(g)) ENCARG(cons_app(map, s)) -> c12(APP(map, s), ENCARG(map)) ENCARG(cons_app(nil, s)) -> c12(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(cons, s)) -> c12(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(filter, s)) -> c12(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(filter2, s)) -> c12(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c12(APP(true, s), ENCARG(true)) ENCARG(cons_app(false, s)) -> c12(APP(false, s), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), s)) -> c12(APP(app(encArg(z0), encArg(z1)), s), ENCARG(cons_app(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, s)) -> c12(APP(minus, s), ENCARG(minus)) ENCARG(cons_app(0, s)) -> c12(APP(0, s), ENCARG(0)) ENCARG(cons_app(s, s)) -> c12(APP(s, s), ENCARG(s)) ENCARG(cons_app(f, s)) -> c12(APP(f, s), ENCARG(f)) ENCARG(cons_app(g, s)) -> c12(APP(g, s), ENCARG(g)) ENCARG(cons_app(map, s)) -> c12(APP(map, s), ENCARG(map)) ENCARG(cons_app(nil, s)) -> c12(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(cons, s)) -> c12(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(filter, s)) -> c12(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(filter2, s)) -> c12(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c12(APP(true, s), ENCARG(true)) ENCARG(cons_app(false, s)) -> c12(APP(false, s), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(cons, s)) -> c12(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(g, s)) -> c12(APP(g, s), ENCARG(g)) ENCARG(cons_app(f, s)) -> c12(APP(f, s), ENCARG(f)) ENCARG(cons_app(false, s)) -> c12(APP(false, s), ENCARG(false)) ENCARG(cons_app(minus, s)) -> c12(APP(minus, s), ENCARG(minus)) ENCARG(cons_app(0, s)) -> c12(APP(0, s), ENCARG(0)) ENCARG(cons_app(filter, s)) -> c12(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(nil, s)) -> c12(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(s, s)) -> c12(APP(s, s), ENCARG(s)) ENCARG(cons_app(map, s)) -> c12(APP(map, s), ENCARG(map)) ENCARG(cons_app(true, s)) -> c12(APP(true, s), ENCARG(true)) ENCARG(cons_app(filter2, s)) -> c12(APP(filter2, s), ENCARG(filter2)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, f)) -> c12(APP(encArg(x0), f), ENCARG(x0)) by ENCARG(cons_app(minus, f)) -> c12(APP(minus, f), ENCARG(minus)) ENCARG(cons_app(0, f)) -> c12(APP(0, f), ENCARG(0)) ENCARG(cons_app(s, f)) -> c12(APP(s, f), ENCARG(s)) ENCARG(cons_app(f, f)) -> c12(APP(f, f), ENCARG(f)) ENCARG(cons_app(g, f)) -> c12(APP(g, f), ENCARG(g)) ENCARG(cons_app(map, f)) -> c12(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c12(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c12(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c12(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c12(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c12(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c12(APP(false, f), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), f)) -> c12(APP(app(encArg(z0), encArg(z1)), f), ENCARG(cons_app(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, f)) -> c12(APP(minus, f), ENCARG(minus)) ENCARG(cons_app(0, f)) -> c12(APP(0, f), ENCARG(0)) ENCARG(cons_app(s, f)) -> c12(APP(s, f), ENCARG(s)) ENCARG(cons_app(f, f)) -> c12(APP(f, f), ENCARG(f)) ENCARG(cons_app(g, f)) -> c12(APP(g, f), ENCARG(g)) ENCARG(cons_app(map, f)) -> c12(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c12(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c12(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c12(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c12(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c12(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c12(APP(false, f), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(map, f)) -> c12(APP(map, f), ENCARG(map)) ENCARG(cons_app(0, f)) -> c12(APP(0, f), ENCARG(0)) ENCARG(cons_app(false, f)) -> c12(APP(false, f), ENCARG(false)) ENCARG(cons_app(s, f)) -> c12(APP(s, f), ENCARG(s)) ENCARG(cons_app(cons, f)) -> c12(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(f, f)) -> c12(APP(f, f), ENCARG(f)) ENCARG(cons_app(nil, f)) -> c12(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(true, f)) -> c12(APP(true, f), ENCARG(true)) ENCARG(cons_app(g, f)) -> c12(APP(g, f), ENCARG(g)) ENCARG(cons_app(filter2, f)) -> c12(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(filter, f)) -> c12(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(minus, f)) -> c12(APP(minus, f), ENCARG(minus)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, g)) -> c12(APP(encArg(x0), g), ENCARG(x0)) by ENCARG(cons_app(minus, g)) -> c12(APP(minus, g), ENCARG(minus)) ENCARG(cons_app(0, g)) -> c12(APP(0, g), ENCARG(0)) ENCARG(cons_app(s, g)) -> c12(APP(s, g), ENCARG(s)) ENCARG(cons_app(f, g)) -> c12(APP(f, g), ENCARG(f)) ENCARG(cons_app(g, g)) -> c12(APP(g, g), ENCARG(g)) ENCARG(cons_app(map, g)) -> c12(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c12(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c12(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c12(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c12(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c12(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c12(APP(false, g), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), g)) -> c12(APP(app(encArg(z0), encArg(z1)), g), ENCARG(cons_app(z0, z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, g)) -> c12(APP(minus, g), ENCARG(minus)) ENCARG(cons_app(0, g)) -> c12(APP(0, g), ENCARG(0)) ENCARG(cons_app(s, g)) -> c12(APP(s, g), ENCARG(s)) ENCARG(cons_app(f, g)) -> c12(APP(f, g), ENCARG(f)) ENCARG(cons_app(g, g)) -> c12(APP(g, g), ENCARG(g)) ENCARG(cons_app(map, g)) -> c12(APP(map, g), ENCARG(map)) ENCARG(cons_app(nil, g)) -> c12(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(cons, g)) -> c12(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(filter, g)) -> c12(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(filter2, g)) -> c12(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(true, g)) -> c12(APP(true, g), ENCARG(true)) ENCARG(cons_app(false, g)) -> c12(APP(false, g), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(s, g)) -> c12(APP(s, g), ENCARG(s)) ENCARG(cons_app(nil, g)) -> c12(APP(nil, g), ENCARG(nil)) ENCARG(cons_app(0, g)) -> c12(APP(0, g), ENCARG(0)) ENCARG(cons_app(cons, g)) -> c12(APP(cons, g), ENCARG(cons)) ENCARG(cons_app(f, g)) -> c12(APP(f, g), ENCARG(f)) ENCARG(cons_app(filter, g)) -> c12(APP(filter, g), ENCARG(filter)) ENCARG(cons_app(minus, g)) -> c12(APP(minus, g), ENCARG(minus)) ENCARG(cons_app(false, g)) -> c12(APP(false, g), ENCARG(false)) ENCARG(cons_app(filter2, g)) -> c12(APP(filter2, g), ENCARG(filter2)) ENCARG(cons_app(g, g)) -> c12(APP(g, g), ENCARG(g)) ENCARG(cons_app(true, g)) -> c12(APP(true, g), ENCARG(true)) ENCARG(cons_app(map, g)) -> c12(APP(map, g), ENCARG(map)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c12(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(minus, map)) -> c12(APP(minus, map), ENCARG(minus)) ENCARG(cons_app(0, map)) -> c12(APP(0, map), ENCARG(0)) ENCARG(cons_app(s, map)) -> c12(APP(s, map), ENCARG(s)) ENCARG(cons_app(f, map)) -> c12(APP(f, map), ENCARG(f)) ENCARG(cons_app(g, map)) -> c12(APP(g, map), ENCARG(g)) ENCARG(cons_app(map, map)) -> c12(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c12(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c12(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c12(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c12(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c12(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c12(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c12(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, map)) -> c12(APP(minus, map), ENCARG(minus)) ENCARG(cons_app(0, map)) -> c12(APP(0, map), ENCARG(0)) ENCARG(cons_app(s, map)) -> c12(APP(s, map), ENCARG(s)) ENCARG(cons_app(f, map)) -> c12(APP(f, map), ENCARG(f)) ENCARG(cons_app(g, map)) -> c12(APP(g, map), ENCARG(g)) ENCARG(cons_app(map, map)) -> c12(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c12(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c12(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c12(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c12(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c12(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c12(APP(false, map), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(nil, map)) -> c12(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(0, map)) -> c12(APP(0, map), ENCARG(0)) ENCARG(cons_app(map, map)) -> c12(APP(map, map), ENCARG(map)) ENCARG(cons_app(filter, map)) -> c12(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(cons, map)) -> c12(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(f, map)) -> c12(APP(f, map), ENCARG(f)) ENCARG(cons_app(minus, map)) -> c12(APP(minus, map), ENCARG(minus)) ENCARG(cons_app(false, map)) -> c12(APP(false, map), ENCARG(false)) ENCARG(cons_app(g, map)) -> c12(APP(g, map), ENCARG(g)) ENCARG(cons_app(true, map)) -> c12(APP(true, map), ENCARG(true)) ENCARG(cons_app(filter2, map)) -> c12(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(s, map)) -> c12(APP(s, map), ENCARG(s)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c12(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(minus, nil)) -> c12(APP(minus, nil), ENCARG(minus)) ENCARG(cons_app(0, nil)) -> c12(APP(0, nil), ENCARG(0)) ENCARG(cons_app(s, nil)) -> c12(APP(s, nil), ENCARG(s)) ENCARG(cons_app(f, nil)) -> c12(APP(f, nil), ENCARG(f)) ENCARG(cons_app(g, nil)) -> c12(APP(g, nil), ENCARG(g)) ENCARG(cons_app(map, nil)) -> c12(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c12(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c12(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c12(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c12(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c12(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c12(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c12(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, nil)) -> c12(APP(minus, nil), ENCARG(minus)) ENCARG(cons_app(0, nil)) -> c12(APP(0, nil), ENCARG(0)) ENCARG(cons_app(s, nil)) -> c12(APP(s, nil), ENCARG(s)) ENCARG(cons_app(f, nil)) -> c12(APP(f, nil), ENCARG(f)) ENCARG(cons_app(g, nil)) -> c12(APP(g, nil), ENCARG(g)) ENCARG(cons_app(map, nil)) -> c12(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c12(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c12(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c12(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c12(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c12(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c12(APP(false, nil), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter2, nil)) -> c12(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(cons, nil)) -> c12(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(map, nil)) -> c12(APP(map, nil), ENCARG(map)) ENCARG(cons_app(f, nil)) -> c12(APP(f, nil), ENCARG(f)) ENCARG(cons_app(minus, nil)) -> c12(APP(minus, nil), ENCARG(minus)) ENCARG(cons_app(0, nil)) -> c12(APP(0, nil), ENCARG(0)) ENCARG(cons_app(nil, nil)) -> c12(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(filter, nil)) -> c12(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(true, nil)) -> c12(APP(true, nil), ENCARG(true)) ENCARG(cons_app(s, nil)) -> c12(APP(s, nil), ENCARG(s)) ENCARG(cons_app(false, nil)) -> c12(APP(false, nil), ENCARG(false)) ENCARG(cons_app(g, nil)) -> c12(APP(g, nil), ENCARG(g)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c12(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(minus, cons)) -> c12(APP(minus, cons), ENCARG(minus)) ENCARG(cons_app(0, cons)) -> c12(APP(0, cons), ENCARG(0)) ENCARG(cons_app(s, cons)) -> c12(APP(s, cons), ENCARG(s)) ENCARG(cons_app(f, cons)) -> c12(APP(f, cons), ENCARG(f)) ENCARG(cons_app(g, cons)) -> c12(APP(g, cons), ENCARG(g)) ENCARG(cons_app(map, cons)) -> c12(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c12(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c12(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c12(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c12(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c12(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c12(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c12(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, cons)) -> c12(APP(minus, cons), ENCARG(minus)) ENCARG(cons_app(0, cons)) -> c12(APP(0, cons), ENCARG(0)) ENCARG(cons_app(s, cons)) -> c12(APP(s, cons), ENCARG(s)) ENCARG(cons_app(f, cons)) -> c12(APP(f, cons), ENCARG(f)) ENCARG(cons_app(g, cons)) -> c12(APP(g, cons), ENCARG(g)) ENCARG(cons_app(map, cons)) -> c12(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c12(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c12(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c12(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c12(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c12(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c12(APP(false, cons), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(0, cons)) -> c12(APP(0, cons), ENCARG(0)) ENCARG(cons_app(g, cons)) -> c12(APP(g, cons), ENCARG(g)) ENCARG(cons_app(minus, cons)) -> c12(APP(minus, cons), ENCARG(minus)) ENCARG(cons_app(false, cons)) -> c12(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons, cons)) -> c12(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(f, cons)) -> c12(APP(f, cons), ENCARG(f)) ENCARG(cons_app(nil, cons)) -> c12(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(true, cons)) -> c12(APP(true, cons), ENCARG(true)) ENCARG(cons_app(filter2, cons)) -> c12(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(s, cons)) -> c12(APP(s, cons), ENCARG(s)) ENCARG(cons_app(filter, cons)) -> c12(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(map, cons)) -> c12(APP(map, cons), ENCARG(map)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c12(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(minus, filter)) -> c12(APP(minus, filter), ENCARG(minus)) ENCARG(cons_app(0, filter)) -> c12(APP(0, filter), ENCARG(0)) ENCARG(cons_app(s, filter)) -> c12(APP(s, filter), ENCARG(s)) ENCARG(cons_app(f, filter)) -> c12(APP(f, filter), ENCARG(f)) ENCARG(cons_app(g, filter)) -> c12(APP(g, filter), ENCARG(g)) ENCARG(cons_app(map, filter)) -> c12(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c12(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c12(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c12(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c12(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c12(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c12(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c12(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, filter)) -> c12(APP(minus, filter), ENCARG(minus)) ENCARG(cons_app(0, filter)) -> c12(APP(0, filter), ENCARG(0)) ENCARG(cons_app(s, filter)) -> c12(APP(s, filter), ENCARG(s)) ENCARG(cons_app(f, filter)) -> c12(APP(f, filter), ENCARG(f)) ENCARG(cons_app(g, filter)) -> c12(APP(g, filter), ENCARG(g)) ENCARG(cons_app(map, filter)) -> c12(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c12(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c12(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c12(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c12(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c12(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c12(APP(false, filter), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(cons, filter)) -> c12(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(nil, filter)) -> c12(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(f, filter)) -> c12(APP(f, filter), ENCARG(f)) ENCARG(cons_app(false, filter)) -> c12(APP(false, filter), ENCARG(false)) ENCARG(cons_app(true, filter)) -> c12(APP(true, filter), ENCARG(true)) ENCARG(cons_app(filter, filter)) -> c12(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(g, filter)) -> c12(APP(g, filter), ENCARG(g)) ENCARG(cons_app(0, filter)) -> c12(APP(0, filter), ENCARG(0)) ENCARG(cons_app(minus, filter)) -> c12(APP(minus, filter), ENCARG(minus)) ENCARG(cons_app(map, filter)) -> c12(APP(map, filter), ENCARG(map)) ENCARG(cons_app(filter2, filter)) -> c12(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(s, filter)) -> c12(APP(s, filter), ENCARG(s)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter2)) -> c12(APP(encArg(x0), filter2), ENCARG(x0)) by ENCARG(cons_app(minus, filter2)) -> c12(APP(minus, filter2), ENCARG(minus)) ENCARG(cons_app(0, filter2)) -> c12(APP(0, filter2), ENCARG(0)) ENCARG(cons_app(s, filter2)) -> c12(APP(s, filter2), ENCARG(s)) ENCARG(cons_app(f, filter2)) -> c12(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(g, filter2)) -> c12(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(map, filter2)) -> c12(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c12(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c12(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c12(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c12(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c12(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c12(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter2)) -> c12(APP(app(encArg(z0), encArg(z1)), filter2), ENCARG(cons_app(z0, z1))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, filter2)) -> c12(APP(minus, filter2), ENCARG(minus)) ENCARG(cons_app(0, filter2)) -> c12(APP(0, filter2), ENCARG(0)) ENCARG(cons_app(s, filter2)) -> c12(APP(s, filter2), ENCARG(s)) ENCARG(cons_app(f, filter2)) -> c12(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(g, filter2)) -> c12(APP(g, filter2), ENCARG(g)) ENCARG(cons_app(map, filter2)) -> c12(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(nil, filter2)) -> c12(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(cons, filter2)) -> c12(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(filter, filter2)) -> c12(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(filter2, filter2)) -> c12(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(true, filter2)) -> c12(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(false, filter2)) -> c12(APP(false, filter2), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(filter2, filter2)) -> c12(APP(filter2, filter2), ENCARG(filter2)) ENCARG(cons_app(s, filter2)) -> c12(APP(s, filter2), ENCARG(s)) ENCARG(cons_app(filter, filter2)) -> c12(APP(filter, filter2), ENCARG(filter)) ENCARG(cons_app(cons, filter2)) -> c12(APP(cons, filter2), ENCARG(cons)) ENCARG(cons_app(f, filter2)) -> c12(APP(f, filter2), ENCARG(f)) ENCARG(cons_app(nil, filter2)) -> c12(APP(nil, filter2), ENCARG(nil)) ENCARG(cons_app(map, filter2)) -> c12(APP(map, filter2), ENCARG(map)) ENCARG(cons_app(0, filter2)) -> c12(APP(0, filter2), ENCARG(0)) ENCARG(cons_app(false, filter2)) -> c12(APP(false, filter2), ENCARG(false)) ENCARG(cons_app(minus, filter2)) -> c12(APP(minus, filter2), ENCARG(minus)) ENCARG(cons_app(true, filter2)) -> c12(APP(true, filter2), ENCARG(true)) ENCARG(cons_app(g, filter2)) -> c12(APP(g, filter2), ENCARG(g)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, true)) -> c12(APP(encArg(x0), true), ENCARG(x0)) by ENCARG(cons_app(minus, true)) -> c12(APP(minus, true), ENCARG(minus)) ENCARG(cons_app(0, true)) -> c12(APP(0, true), ENCARG(0)) ENCARG(cons_app(s, true)) -> c12(APP(s, true), ENCARG(s)) ENCARG(cons_app(f, true)) -> c12(APP(f, true), ENCARG(f)) ENCARG(cons_app(g, true)) -> c12(APP(g, true), ENCARG(g)) ENCARG(cons_app(map, true)) -> c12(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c12(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c12(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c12(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c12(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c12(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c12(APP(false, true), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), true)) -> c12(APP(app(encArg(z0), encArg(z1)), true), ENCARG(cons_app(z0, z1))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, true)) -> c12(APP(minus, true), ENCARG(minus)) ENCARG(cons_app(0, true)) -> c12(APP(0, true), ENCARG(0)) ENCARG(cons_app(s, true)) -> c12(APP(s, true), ENCARG(s)) ENCARG(cons_app(f, true)) -> c12(APP(f, true), ENCARG(f)) ENCARG(cons_app(g, true)) -> c12(APP(g, true), ENCARG(g)) ENCARG(cons_app(map, true)) -> c12(APP(map, true), ENCARG(map)) ENCARG(cons_app(nil, true)) -> c12(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(cons, true)) -> c12(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(filter, true)) -> c12(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(filter2, true)) -> c12(APP(filter2, true), ENCARG(filter2)) ENCARG(cons_app(true, true)) -> c12(APP(true, true), ENCARG(true)) ENCARG(cons_app(false, true)) -> c12(APP(false, true), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(cons, true)) -> c12(APP(cons, true), ENCARG(cons)) ENCARG(cons_app(f, true)) -> c12(APP(f, true), ENCARG(f)) ENCARG(cons_app(minus, true)) -> c12(APP(minus, true), ENCARG(minus)) ENCARG(cons_app(0, true)) -> c12(APP(0, true), ENCARG(0)) ENCARG(cons_app(false, true)) -> c12(APP(false, true), ENCARG(false)) ENCARG(cons_app(true, true)) -> c12(APP(true, true), ENCARG(true)) ENCARG(cons_app(s, true)) -> c12(APP(s, true), ENCARG(s)) ENCARG(cons_app(nil, true)) -> c12(APP(nil, true), ENCARG(nil)) ENCARG(cons_app(g, true)) -> c12(APP(g, true), ENCARG(g)) ENCARG(cons_app(filter, true)) -> c12(APP(filter, true), ENCARG(filter)) ENCARG(cons_app(map, true)) -> c12(APP(map, true), ENCARG(map)) ENCARG(cons_app(filter2, true)) -> c12(APP(filter2, true), ENCARG(filter2)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_2, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, false)) -> c12(APP(encArg(x0), false), ENCARG(x0)) by ENCARG(cons_app(minus, false)) -> c12(APP(minus, false), ENCARG(minus)) ENCARG(cons_app(0, false)) -> c12(APP(0, false), ENCARG(0)) ENCARG(cons_app(s, false)) -> c12(APP(s, false), ENCARG(s)) ENCARG(cons_app(f, false)) -> c12(APP(f, false), ENCARG(f)) ENCARG(cons_app(g, false)) -> c12(APP(g, false), ENCARG(g)) ENCARG(cons_app(map, false)) -> c12(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c12(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c12(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c12(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c12(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c12(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c12(APP(false, false), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), false)) -> c12(APP(app(encArg(z0), encArg(z1)), false), ENCARG(cons_app(z0, z1))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(minus, false)) -> c12(APP(minus, false), ENCARG(minus)) ENCARG(cons_app(0, false)) -> c12(APP(0, false), ENCARG(0)) ENCARG(cons_app(s, false)) -> c12(APP(s, false), ENCARG(s)) ENCARG(cons_app(f, false)) -> c12(APP(f, false), ENCARG(f)) ENCARG(cons_app(g, false)) -> c12(APP(g, false), ENCARG(g)) ENCARG(cons_app(map, false)) -> c12(APP(map, false), ENCARG(map)) ENCARG(cons_app(nil, false)) -> c12(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(cons, false)) -> c12(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(filter, false)) -> c12(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(filter2, false)) -> c12(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(true, false)) -> c12(APP(true, false), ENCARG(true)) ENCARG(cons_app(false, false)) -> c12(APP(false, false), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c12_2, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: ENCARG(cons_app(s, false)) -> c12(APP(s, false), ENCARG(s)) ENCARG(cons_app(g, false)) -> c12(APP(g, false), ENCARG(g)) ENCARG(cons_app(map, false)) -> c12(APP(map, false), ENCARG(map)) ENCARG(cons_app(false, false)) -> c12(APP(false, false), ENCARG(false)) ENCARG(cons_app(filter2, false)) -> c12(APP(filter2, false), ENCARG(filter2)) ENCARG(cons_app(nil, false)) -> c12(APP(nil, false), ENCARG(nil)) ENCARG(cons_app(0, false)) -> c12(APP(0, false), ENCARG(0)) ENCARG(cons_app(minus, false)) -> c12(APP(minus, false), ENCARG(minus)) ENCARG(cons_app(filter, false)) -> c12(APP(filter, false), ENCARG(filter)) ENCARG(cons_app(true, false)) -> c12(APP(true, false), ENCARG(true)) ENCARG(cons_app(cons, false)) -> c12(APP(cons, false), ENCARG(cons)) ENCARG(cons_app(f, false)) -> c12(APP(f, false), ENCARG(f)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c12_2, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(f, x1)) -> c12(APP(f, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(f, minus)) -> c12(APP(f, minus), ENCARG(minus)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0), ENCARG(0)) ENCARG(cons_app(f, s)) -> c12(APP(f, s), ENCARG(s)) ENCARG(cons_app(f, f)) -> c12(APP(f, f), ENCARG(f)) ENCARG(cons_app(f, g)) -> c12(APP(f, g), ENCARG(g)) ENCARG(cons_app(f, map)) -> c12(APP(f, map), ENCARG(map)) ENCARG(cons_app(f, nil)) -> c12(APP(f, nil), ENCARG(nil)) ENCARG(cons_app(f, cons)) -> c12(APP(f, cons), ENCARG(cons)) ENCARG(cons_app(f, filter)) -> c12(APP(f, filter), ENCARG(filter)) ENCARG(cons_app(f, filter2)) -> c12(APP(f, filter2), ENCARG(filter2)) ENCARG(cons_app(f, true)) -> c12(APP(f, true), ENCARG(true)) ENCARG(cons_app(f, false)) -> c12(APP(f, false), ENCARG(false)) ENCARG(cons_app(f, cons_app(z0, z1))) -> c12(APP(f, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(f, minus)) -> c12(APP(f, minus), ENCARG(minus)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0), ENCARG(0)) ENCARG(cons_app(f, s)) -> c12(APP(f, s), ENCARG(s)) ENCARG(cons_app(f, f)) -> c12(APP(f, f), ENCARG(f)) ENCARG(cons_app(f, g)) -> c12(APP(f, g), ENCARG(g)) ENCARG(cons_app(f, map)) -> c12(APP(f, map), ENCARG(map)) ENCARG(cons_app(f, nil)) -> c12(APP(f, nil), ENCARG(nil)) ENCARG(cons_app(f, cons)) -> c12(APP(f, cons), ENCARG(cons)) ENCARG(cons_app(f, filter)) -> c12(APP(f, filter), ENCARG(filter)) ENCARG(cons_app(f, filter2)) -> c12(APP(f, filter2), ENCARG(filter2)) ENCARG(cons_app(f, true)) -> c12(APP(f, true), ENCARG(true)) ENCARG(cons_app(f, false)) -> c12(APP(f, false), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c12_2, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(f, filter)) -> c12(APP(f, filter), ENCARG(filter)) ENCARG(cons_app(f, minus)) -> c12(APP(f, minus), ENCARG(minus)) ENCARG(cons_app(f, nil)) -> c12(APP(f, nil), ENCARG(nil)) ENCARG(cons_app(f, false)) -> c12(APP(f, false), ENCARG(false)) ENCARG(cons_app(f, true)) -> c12(APP(f, true), ENCARG(true)) ENCARG(cons_app(f, cons)) -> c12(APP(f, cons), ENCARG(cons)) ENCARG(cons_app(f, s)) -> c12(APP(f, s), ENCARG(s)) ENCARG(cons_app(f, filter2)) -> c12(APP(f, filter2), ENCARG(filter2)) ENCARG(cons_app(f, f)) -> c12(APP(f, f), ENCARG(f)) ENCARG(cons_app(f, g)) -> c12(APP(f, g), ENCARG(g)) ENCARG(cons_app(f, map)) -> c12(APP(f, map), ENCARG(map)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0), ENCARG(0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c12_2, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c12_2, c29_3, c31_3, c35_2, c35_1, c_1, c12_3 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(g, x1)) -> c12(APP(g, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(g, minus)) -> c12(APP(g, minus), ENCARG(minus)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0), ENCARG(0)) ENCARG(cons_app(g, s)) -> c12(APP(g, s), ENCARG(s)) ENCARG(cons_app(g, f)) -> c12(APP(g, f), ENCARG(f)) ENCARG(cons_app(g, g)) -> c12(APP(g, g), ENCARG(g)) ENCARG(cons_app(g, map)) -> c12(APP(g, map), ENCARG(map)) ENCARG(cons_app(g, nil)) -> c12(APP(g, nil), ENCARG(nil)) ENCARG(cons_app(g, cons)) -> c12(APP(g, cons), ENCARG(cons)) ENCARG(cons_app(g, filter)) -> c12(APP(g, filter), ENCARG(filter)) ENCARG(cons_app(g, filter2)) -> c12(APP(g, filter2), ENCARG(filter2)) ENCARG(cons_app(g, true)) -> c12(APP(g, true), ENCARG(true)) ENCARG(cons_app(g, false)) -> c12(APP(g, false), ENCARG(false)) ENCARG(cons_app(g, cons_app(z0, z1))) -> c12(APP(g, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(g, minus)) -> c12(APP(g, minus), ENCARG(minus)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0), ENCARG(0)) ENCARG(cons_app(g, s)) -> c12(APP(g, s), ENCARG(s)) ENCARG(cons_app(g, f)) -> c12(APP(g, f), ENCARG(f)) ENCARG(cons_app(g, g)) -> c12(APP(g, g), ENCARG(g)) ENCARG(cons_app(g, map)) -> c12(APP(g, map), ENCARG(map)) ENCARG(cons_app(g, nil)) -> c12(APP(g, nil), ENCARG(nil)) ENCARG(cons_app(g, cons)) -> c12(APP(g, cons), ENCARG(cons)) ENCARG(cons_app(g, filter)) -> c12(APP(g, filter), ENCARG(filter)) ENCARG(cons_app(g, filter2)) -> c12(APP(g, filter2), ENCARG(filter2)) ENCARG(cons_app(g, true)) -> c12(APP(g, true), ENCARG(true)) ENCARG(cons_app(g, false)) -> c12(APP(g, false), ENCARG(false)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3, c12_2 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(g, map)) -> c12(APP(g, map), ENCARG(map)) ENCARG(cons_app(g, false)) -> c12(APP(g, false), ENCARG(false)) ENCARG(cons_app(g, true)) -> c12(APP(g, true), ENCARG(true)) ENCARG(cons_app(g, minus)) -> c12(APP(g, minus), ENCARG(minus)) ENCARG(cons_app(g, g)) -> c12(APP(g, g), ENCARG(g)) ENCARG(cons_app(g, cons)) -> c12(APP(g, cons), ENCARG(cons)) ENCARG(cons_app(g, f)) -> c12(APP(g, f), ENCARG(f)) ENCARG(cons_app(g, filter2)) -> c12(APP(g, filter2), ENCARG(filter2)) ENCARG(cons_app(g, filter)) -> c12(APP(g, filter), ENCARG(filter)) ENCARG(cons_app(g, nil)) -> c12(APP(g, nil), ENCARG(nil)) ENCARG(cons_app(g, s)) -> c12(APP(g, s), ENCARG(s)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0), ENCARG(0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3, c12_2 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3, c12_2 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(g, app(s, 0))), APP(g, app(f, 0)), APP(f, 0)) by APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c29_3, c31_3, c35_2, c35_1, c_1, c12_3, c12_2, c29_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(f, app(s, app(s, z0))) -> c29(APP(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))), APP(g, app(f, app(s, z0))), APP(f, app(s, z0))) by APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c31_3, c35_2, c35_1, c_1, c12_3, c12_2, c29_3, c29_2, c29_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(f, 0)), APP(f, app(g, 0)), APP(g, 0)) by APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c31_3, c35_2, c35_1, c_1, c12_3, c12_2, c29_3, c29_2, c29_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(g, app(s, app(s, z0))) -> c31(APP(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))), APP(f, app(g, app(s, z0))), APP(g, app(s, z0))) by APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c35_2, c35_1, c_1, c12_3, c12_2, c29_3, c29_2, c29_1, c31_3, c31_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, z0))), app(s, z1)), x2), APP(app(minus, app(s, z0)), app(s, z1))) by APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c35_2, c35_1, c_1, c12_3, c12_2, c29_3, c29_2, c29_1, c31_3, c31_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, f), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(g, app(f, z0)))), f), app(s, z0)), x2), APP(f, app(s, z0))) by APP(app(filter, f), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(g, app(s, 0)))), f), app(s, 0)), x1), APP(f, app(s, 0))) APP(app(filter, f), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0)))))), f), app(s, app(s, z0))), x1), APP(f, app(s, app(s, z0)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) APP(app(filter, f), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(g, app(s, 0)))), f), app(s, 0)), x1), APP(f, app(s, 0))) APP(app(filter, f), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0)))))), f), app(s, app(s, z0))), x1), APP(f, app(s, app(s, z0)))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) APP(app(filter, f), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(g, app(s, 0)))), f), app(s, 0)), x1), APP(f, app(s, 0))) APP(app(filter, f), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0)))))), f), app(s, app(s, z0))), x1), APP(f, app(s, app(s, z0)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c35_2, c35_1, c_1, c12_3, c12_2, c29_3, c29_2, c29_1, c31_3, c31_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, g), app(app(cons, app(s, z0)), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, z0)), app(f, app(g, z0)))), g), app(s, z0)), x2), APP(g, app(s, z0))) by APP(app(filter, g), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(f, 0))), g), app(s, 0)), x1), APP(g, app(s, 0))) APP(app(filter, g), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0)))))), g), app(s, app(s, z0))), x1), APP(g, app(s, app(s, z0)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: encArg(minus) -> minus encArg(0) -> 0 encArg(s) -> s encArg(f) -> f encArg(g) -> g encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(minus, z0), 0) -> z0 app(app(minus, app(s, z0)), app(s, z1)) -> app(app(minus, z0), z1) app(f, 0) -> app(s, 0) app(f, app(s, z0)) -> app(app(minus, app(s, z0)), app(g, app(f, z0))) app(g, 0) -> 0 app(g, app(s, z0)) -> app(app(minus, app(s, z0)), app(f, app(g, 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(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) ENCARG(cons_app(minus, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c12(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c12(ENCARG(x1)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, minus) -> c(APP(encArg(x0), minus)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, g) -> c(APP(encArg(x0), g)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(g, x1) -> c(APP(g, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, minus))) -> c12(APP(encArg(x0), app(encArg(x1), minus)), ENCARG(x0), ENCARG(cons_app(x1, minus))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c12(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c12(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c12(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, g))) -> c12(APP(encArg(x0), app(encArg(x1), g)), ENCARG(x0), ENCARG(cons_app(x1, g))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c12(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c12(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c12(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c12(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c12(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c12(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c12(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c12(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(minus, x2))) -> c12(APP(encArg(x0), app(minus, encArg(x2))), ENCARG(x0), ENCARG(cons_app(minus, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c12(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c12(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c12(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(g, x2))) -> c12(APP(encArg(x0), app(g, encArg(x2))), ENCARG(x0), ENCARG(cons_app(g, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c12(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c12(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c12(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c12(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c12(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c12(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c12(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c12(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c12(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(minus, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c12(APP(f, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(g, cons_app(x1, x2))) -> c12(APP(g, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c12(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, minus), x2)) -> c12(APP(app(encArg(x0), minus), encArg(x2)), ENCARG(cons_app(x0, minus)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c12(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c12(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c12(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, g), x2)) -> c12(APP(app(encArg(x0), g), encArg(x2)), ENCARG(cons_app(x0, g)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c12(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c12(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c12(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c12(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c12(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c12(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c12(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c12(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(minus, x1), x2)) -> c12(APP(app(minus, encArg(x1)), encArg(x2)), ENCARG(cons_app(minus, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c12(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(g, x1), x2)) -> c12(APP(app(g, encArg(x1)), encArg(x2)), ENCARG(cons_app(g, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c12(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c12(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c12(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), minus)) -> c12(APP(app(encArg(x0), encArg(x1)), minus), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c12(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c12(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c12(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), g)) -> c12(APP(app(encArg(x0), encArg(x1)), g), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c12(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c12(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c12(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c12(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c12(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c12(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c12(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c12(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c12(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c12(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c12(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c12(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c12(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c12(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c12(APP(f, 0)) ENCARG(cons_app(g, 0)) -> c12(APP(g, 0)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) APP(app(filter, f), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(g, app(s, 0)))), f), app(s, 0)), x1), APP(f, app(s, 0))) APP(app(filter, f), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0)))))), f), app(s, app(s, z0))), x1), APP(f, app(s, app(s, z0)))) APP(app(filter, g), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(f, 0))), g), app(s, 0)), x1), APP(g, app(s, 0))) APP(app(filter, g), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0)))))), g), app(s, app(s, z0))), x1), APP(g, app(s, app(s, z0)))) S tuples: APP(app(minus, z0), 0) -> c26 APP(g, 0) -> c30 APP(app(map, z0), nil) -> c32 APP(app(filter, z0), nil) -> c34 APP(app(minus, app(s, z0)), app(s, z1)) -> c27(APP(app(minus, z0), z1)) APP(f, 0) -> c28 APP(app(map, z0), app(app(cons, z1), z2)) -> c33(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c36(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c37(APP(app(filter, z0), z2)) APP(app(filter, app(minus, z0)), app(app(cons, 0), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, z0)), 0), x2), APP(app(minus, z0), 0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, f), app(app(cons, 0), x2)) -> c35(APP(f, 0)) APP(app(filter, g), app(app(cons, 0), x2)) -> c35(APP(g, 0)) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c35(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c35(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c35(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c35(APP(app(app(app(filter2, true), z0), z1), z2)) APP(f, app(s, 0)) -> c29(APP(app(minus, app(s, 0)), app(app(minus, app(s, 0)), app(f, app(g, 0)))), APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, 0)) -> c29(APP(g, app(f, 0)), APP(f, 0)) APP(f, app(s, app(s, 0))) -> c29(APP(app(minus, app(s, app(s, 0))), app(g, app(app(minus, app(s, 0)), app(g, app(s, 0))))), APP(g, app(f, app(s, 0))), APP(f, app(s, 0))) APP(f, app(s, app(s, app(s, z0)))) -> c29(APP(app(minus, app(s, app(s, app(s, z0)))), app(g, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0))))))), APP(g, app(f, app(s, app(s, z0)))), APP(f, app(s, app(s, z0)))) APP(f, app(s, app(s, x0))) -> c29(APP(g, app(f, app(s, x0)))) APP(g, app(s, 0)) -> c31(APP(app(minus, app(s, 0)), app(s, 0)), APP(f, app(g, 0)), APP(g, 0)) APP(g, app(s, app(s, 0))) -> c31(APP(app(minus, app(s, app(s, 0))), app(f, app(app(minus, app(s, 0)), app(f, 0)))), APP(f, app(g, app(s, 0))), APP(g, app(s, 0))) APP(g, app(s, app(s, app(s, z0)))) -> c31(APP(app(minus, app(s, app(s, app(s, z0)))), app(f, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0))))))), APP(f, app(g, app(s, app(s, z0)))), APP(g, app(s, app(s, z0)))) APP(g, app(s, app(s, x0))) -> c31(APP(f, app(g, app(s, x0)))) APP(app(filter, app(minus, app(s, z0))), app(app(cons, app(s, 0)), x2)) -> c35(APP(app(app(app(filter2, z0), app(minus, app(s, z0))), app(s, 0)), x2), APP(app(minus, app(s, z0)), app(s, 0))) APP(app(filter, app(minus, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c35(APP(app(app(app(filter2, app(app(minus, z0), z1)), app(minus, app(s, app(s, z0)))), app(s, app(s, z1))), x2), APP(app(minus, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(minus, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c35(APP(app(minus, app(s, x0)), app(s, x1))) APP(app(filter, f), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(g, app(s, 0)))), f), app(s, 0)), x1), APP(f, app(s, 0))) APP(app(filter, f), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(g, app(app(minus, app(s, z0)), app(g, app(f, z0)))))), f), app(s, app(s, z0))), x1), APP(f, app(s, app(s, z0)))) APP(app(filter, g), app(app(cons, app(s, 0)), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, 0)), app(f, 0))), g), app(s, 0)), x1), APP(g, app(s, 0))) APP(app(filter, g), app(app(cons, app(s, app(s, z0))), x1)) -> c35(APP(app(app(app(filter2, app(app(minus, app(s, app(s, z0))), app(f, app(app(minus, app(s, z0)), app(f, app(g, z0)))))), g), app(s, app(s, z0))), x1), APP(g, app(s, app(s, z0)))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c26, c30, c32, c34, c27_1, c28, c33_2, c36_1, c37_1, c12_1, c35_2, c35_1, c_1, c12_3, c12_2, c29_3, c29_2, c29_1, c31_3, c31_1