/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 196 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), 7596 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 33 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), 14 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), 3 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 11 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 59 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 82 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 19 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 26 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 33 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 23 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(app(f, 0'), 1'), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0'), 1'), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(0') -> 0' encArg(1') -> 1' encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0' encode_1 -> 1' encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(app(app(f, 0'), 1'), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0'), 1'), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) encArg(f) -> f encArg(0') -> 0' encArg(1') -> 1' encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0' encode_1 -> 1' encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app f :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app 0' :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app 1' :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app s :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app map :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app nil :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app cons :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app filter :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app true :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app false :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encArg :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_0 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_1 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_s :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app hole_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: app(app(app(f, 0'), 1'), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0'), 1'), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) encArg(f) -> f encArg(0') -> 0' encArg(1') -> 1' encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0' encode_1 -> 1' encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false Types: app :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app f :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app 0' :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app 1' :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app s :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app map :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app nil :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app cons :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app filter :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app filter2 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app true :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app false :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encArg :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app cons_app :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_app :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_f :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_0 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_1 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_s :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_map :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_nil :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_cons :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_filter :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_filter2 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_true :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app encode_false :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app hole_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app1_3 :: f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3 :: Nat -> f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app Generator Equations: gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(0) <=> f gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(x, 1)) <=> cons_app(f, gen_f:0':1':s: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_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(n1530_3)) -> *3_3, rt in Omega(0) Induction Base: encArg(gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(0)) Induction Step: encArg(gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(+(n1530_3, 1))) ->_R^Omega(0) app(encArg(f), encArg(gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(n1530_3))) ->_R^Omega(0) app(f, encArg(gen_f:0':1':s:map:nil:cons:filter:filter2:true:false:cons_app2_3(n1530_3))) ->_IH app(f, *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(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) app(app(map, fun), nil) -> nil app(app(map, fun), app(app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(app(filter, fun), nil) -> nil app(app(filter, fun), app(app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(app(app(app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(app(app(app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) app(c_app(map, fun), nil) -> nil app(c_app(filter, fun), nil) -> nil app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) app(c_app(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) The (relative) TRS S consists of the following rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(x0, x1) -> c_app(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) [1] app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) [1] encArg(f) -> f [0] encArg(0) -> 0 [0] encArg(1) -> 1 [0] encArg(s) -> s [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_f -> f [0] encode_0 -> 0 [0] encode_1 -> 1 [0] encode_s -> s [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) [1] app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) [1] encArg(f) -> f [0] encArg(0) -> 0 [0] encArg(1) -> 1 [0] encArg(s) -> s [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_f -> f [0] encode_0 -> 0 [0] encode_1 -> 1 [0] encode_s -> s [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] The TRS has the following type information: app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app c_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app f :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app 0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app 1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app cons_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_f :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1: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_f encode_0 encode_1 encode_s encode_map encode_nil encode_cons encode_filter encode_filter2 encode_true encode_false app_2 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_f -> null_encode_f [0] encode_0 -> null_encode_0 [0] encode_1 -> null_encode_1 [0] encode_s -> null_encode_s [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_f, null_encode_0, null_encode_1, null_encode_s, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) [1] app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) [1] encArg(f) -> f [0] encArg(0) -> 0 [0] encArg(1) -> 1 [0] encArg(s) -> s [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_f -> f [0] encode_0 -> 0 [0] encode_1 -> 1 [0] encode_s -> s [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_f -> null_encode_f [0] encode_0 -> null_encode_0 [0] encode_1 -> null_encode_1 [0] encode_s -> null_encode_s [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(map, c_app(map, fun')), c_app(c_app(cons, c_app(c_app(cons, x'), xs')), xs)) -> app(app(cons, app(app(cons, app(fun', x')), app(app(map, fun'), xs'))), app(c_app(map, c_app(map, fun')), xs)) [2] app(c_app(map, c_app(map, fun')), c_app(c_app(cons, c_app(c_app(cons, x'), xs')), xs)) -> app(app(cons, app(app(cons, app(fun', x')), app(app(map, fun'), xs'))), app(null_app, xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, true), fun''), x'')), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(cons, x''), app(app(filter, fun''), x))), app(c_app(map, c_app(c_app(c_app(filter2, true), fun''), x'')), xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, true), fun''), x'')), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(cons, x''), app(app(filter, fun''), x))), app(null_app, xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, false), fun1), x2)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(filter, fun1), x)), app(c_app(map, c_app(c_app(c_app(filter2, false), fun1), x2)), xs)) [2] app(c_app(map, c_app(c_app(c_app(filter2, false), fun1), x2)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(filter, fun1), x)), app(null_app, xs)) [2] app(c_app(map, c_app(map, fun2)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(c_app(map, c_app(map, fun2)), xs)) [2] app(c_app(map, c_app(map, fun2)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(null_app, xs)) [2] app(c_app(map, c_app(filter, fun3)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(c_app(map, c_app(filter, fun3)), xs)) [2] app(c_app(map, c_app(filter, fun3)), c_app(c_app(cons, nil), xs)) -> app(app(cons, nil), app(null_app, xs)) [2] app(c_app(map, c_app(filter, fun4)), c_app(c_app(cons, c_app(c_app(cons, x3), xs'')), xs)) -> app(app(cons, app(app(app(app(filter2, app(fun4, x3)), fun4), x3), xs'')), app(c_app(map, c_app(filter, fun4)), xs)) [2] app(c_app(map, c_app(filter, fun4)), c_app(c_app(cons, c_app(c_app(cons, x3), xs'')), xs)) -> app(app(cons, app(app(app(app(filter2, app(fun4, x3)), fun4), x3), xs'')), app(null_app, xs)) [2] app(c_app(map, c_app(c_app(f, x4), y')), c_app(c_app(cons, c_app(s, z')), xs)) -> app(app(cons, app(s, app(app(app(f, 0), 1), z'))), app(c_app(map, c_app(c_app(f, x4), y')), xs)) [2] app(c_app(map, c_app(c_app(f, x4), y')), c_app(c_app(cons, c_app(s, z')), xs)) -> app(app(cons, app(s, app(app(app(f, 0), 1), z'))), app(null_app, xs)) [2] app(c_app(map, c_app(c_app(f, 0), 1)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(app(f, app(s, x)), x), x)), app(c_app(map, c_app(c_app(f, 0), 1)), xs)) [2] app(c_app(map, c_app(c_app(f, 0), 1)), c_app(c_app(cons, x), xs)) -> app(app(cons, app(app(app(f, app(s, x)), x), x)), app(null_app, xs)) [2] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, c_app(fun, x)), app(c_app(map, fun), xs)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, c_app(fun, x)), app(null_app, xs)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, null_app), app(c_app(map, fun), xs)) [1] app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, null_app), app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(c_app(cons, x), app(c_app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(c_app(cons, x), app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(null_app, app(c_app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(null_app, app(null_app, xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(c_app(filter, fun), xs) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(null_app, xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, c_app(map, fun5)), c_app(c_app(cons, c_app(c_app(cons, x5), xs1)), xs)) -> app(app(app(app(filter2, app(app(cons, app(fun5, x5)), app(app(map, fun5), xs1))), c_app(map, fun5)), c_app(c_app(cons, x5), xs1)), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, true), fun6), x6)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(cons, x6), app(app(filter, fun6), x))), c_app(c_app(c_app(filter2, true), fun6), x6)), x), xs) [2] app(c_app(filter, c_app(c_app(c_app(filter2, false), fun7), x7)), 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), x7)), 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, x8), xs2)), xs)) -> app(app(app(app(filter2, app(app(app(app(filter2, app(fun10, x8)), fun10), x8), xs2)), c_app(filter, fun10)), c_app(c_app(cons, x8), xs2)), xs) [2] app(c_app(filter, c_app(c_app(f, x9), y'')), c_app(c_app(cons, c_app(s, z'')), xs)) -> app(app(app(app(filter2, app(s, app(app(app(f, 0), 1), z''))), c_app(c_app(f, x9), y'')), c_app(s, z'')), xs) [2] app(c_app(filter, c_app(c_app(f, 0), 1)), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(app(app(f, app(s, x)), x), x)), c_app(c_app(f, 0), 1)), x), 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(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(c_app(f, 0), 1), z)) [1] app(c_app(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(null_app, 1), z)) [1] app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, c_app(s, x)), x), x) [1] app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, null_app), x), x) [1] encArg(f) -> f [0] encArg(0) -> 0 [0] encArg(1) -> 1 [0] encArg(s) -> s [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_f -> f [0] encode_0 -> 0 [0] encode_1 -> 1 [0] encode_s -> s [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_f -> null_encode_f [0] encode_0 -> null_encode_0 [0] encode_1 -> null_encode_1 [0] encode_s -> null_encode_s [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: map => 7 cons => 2 filter2 => 6 true => 10 filter => 5 false => 4 nil => 8 f => 3 s => 9 0 => 0 1 => 1 null_encArg => 0 null_encode_app => 0 null_encode_f => 0 null_encode_0 => 0 null_encode_1 => 0 null_encode_s => 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(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(app(app(6, app(fun10, x8)), fun10), x8), xs2)), 1 + 5 + fun10), 1 + (1 + 2 + x8) + xs2), xs) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x8) + xs2)) + xs, xs >= 0, xs2 >= 0, z1 = 1 + 5 + (1 + 5 + fun10), x8 >= 0, fun10 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(app(3, app(9, x)), x), x)), 1 + (1 + 3 + 0) + 1), x), xs) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 5 + (1 + (1 + 3 + 0) + 1) app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(5, fun7), x)), 1 + (1 + (1 + 6 + 4) + fun7) + x7), x), xs) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x7 >= 0, x >= 0, z1 = 1 + 5 + (1 + (1 + (1 + 6 + 4) + fun7) + x7), fun7 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(2, x6), app(app(5, fun6), x))), 1 + (1 + (1 + 6 + 10) + fun6) + x6), x), xs) :|: fun6 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, x6 >= 0, z1 = 1 + 5 + (1 + (1 + (1 + 6 + 10) + fun6) + x6) app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(2, app(fun5, x5)), app(app(7, fun5), xs1))), 1 + 7 + fun5), 1 + (1 + 2 + x5) + xs1), xs) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x5) + xs1)) + xs, xs >= 0, x5 >= 0, z1 = 1 + 5 + (1 + 7 + fun5), xs1 >= 0, fun5 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, app(9, app(app(app(3, 0), 1), z''))), 1 + (1 + 3 + x9) + y''), 1 + 9 + z''), xs) :|: xs >= 0, z'' >= 0, z1 = 1 + 5 + (1 + (1 + 3 + x9) + y''), z2 = 1 + (1 + 2 + (1 + 9 + z'')) + xs, y'' >= 0, x9 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, 8), 1 + 7 + fun8), 8), xs) :|: z2 = 1 + (1 + 2 + 8) + xs, xs >= 0, z1 = 1 + 5 + (1 + 7 + fun8), fun8 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, 8), 1 + 5 + fun9), 8), xs) :|: z2 = 1 + (1 + 2 + 8) + xs, xs >= 0, z1 = 1 + 5 + (1 + 5 + fun9), fun9 >= 0 app(z1, z2) -{ 1 }-> app(app(app(app(6, 0), fun), x), xs) :|: fun >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 5 + fun app(z1, z2) -{ 1 }-> app(app(app(app(6, 1 + fun + x), fun), x), xs) :|: fun >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 5 + fun app(z1, z2) -{ 1 }-> app(app(app(3, 0), x), x) :|: z2 = x, x >= 0, z1 = 1 + (1 + 3 + 0) + 1 app(z1, z2) -{ 1 }-> app(app(app(3, 1 + 9 + x), x), x) :|: z2 = x, x >= 0, z1 = 1 + (1 + 3 + 0) + 1 app(z1, z2) -{ 2 }-> app(app(2, app(app(app(app(6, app(fun4, x3)), fun4), x3), xs'')), app(0, xs)) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x3) + xs'')) + xs, xs >= 0, z1 = 1 + 7 + (1 + 5 + fun4), xs'' >= 0, fun4 >= 0, x3 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(app(app(6, app(fun4, x3)), fun4), x3), xs'')), app(1 + 7 + (1 + 5 + fun4), xs)) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x3) + xs'')) + xs, xs >= 0, z1 = 1 + 7 + (1 + 5 + fun4), xs'' >= 0, fun4 >= 0, x3 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(app(3, app(9, x)), x), x)), app(0, xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + 3 + 0) + 1) app(z1, z2) -{ 2 }-> app(app(2, app(app(app(3, app(9, x)), x), x)), app(1 + 7 + (1 + (1 + 3 + 0) + 1), xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + 3 + 0) + 1) app(z1, z2) -{ 2 }-> app(app(2, app(app(5, fun1), x)), app(0, xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 4) + fun1) + x2), fun1 >= 0, x2 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(5, fun1), x)), app(1 + 7 + (1 + (1 + (1 + 6 + 4) + fun1) + x2), xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 4) + fun1) + x2), fun1 >= 0, x2 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, x''), app(app(5, fun''), x))), app(0, xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 10) + fun'') + x''), x >= 0, fun'' >= 0, x'' >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, x''), app(app(5, fun''), x))), app(1 + 7 + (1 + (1 + (1 + 6 + 10) + fun'') + x''), xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 10) + fun'') + x''), x >= 0, fun'' >= 0, x'' >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, app(fun', x')), app(app(7, fun'), xs'))), app(0, xs)) :|: xs >= 0, z1 = 1 + 7 + (1 + 7 + fun'), x' >= 0, xs' >= 0, z2 = 1 + (1 + 2 + (1 + (1 + 2 + x') + xs')) + xs, fun' >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, app(fun', x')), app(app(7, fun'), xs'))), app(1 + 7 + (1 + 7 + fun'), xs)) :|: xs >= 0, z1 = 1 + 7 + (1 + 7 + fun'), x' >= 0, xs' >= 0, z2 = 1 + (1 + 2 + (1 + (1 + 2 + x') + xs')) + xs, fun' >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(9, app(app(app(3, 0), 1), z'))), app(0, xs)) :|: xs >= 0, x4 >= 0, z' >= 0, y' >= 0, z2 = 1 + (1 + 2 + (1 + 9 + z')) + xs, z1 = 1 + 7 + (1 + (1 + 3 + x4) + y') app(z1, z2) -{ 2 }-> app(app(2, app(9, app(app(app(3, 0), 1), z'))), app(1 + 7 + (1 + (1 + 3 + x4) + y'), xs)) :|: xs >= 0, x4 >= 0, z' >= 0, y' >= 0, z2 = 1 + (1 + 2 + (1 + 9 + z')) + xs, z1 = 1 + 7 + (1 + (1 + 3 + x4) + y') app(z1, z2) -{ 2 }-> app(app(2, 8), app(0, xs)) :|: z2 = 1 + (1 + 2 + 8) + xs, fun2 >= 0, xs >= 0, z1 = 1 + 7 + (1 + 7 + fun2) app(z1, z2) -{ 2 }-> app(app(2, 8), app(0, xs)) :|: z2 = 1 + (1 + 2 + 8) + xs, xs >= 0, z1 = 1 + 7 + (1 + 5 + fun3), fun3 >= 0 app(z1, z2) -{ 2 }-> app(app(2, 8), app(1 + 7 + (1 + 7 + fun2), xs)) :|: z2 = 1 + (1 + 2 + 8) + xs, fun2 >= 0, xs >= 0, z1 = 1 + 7 + (1 + 7 + fun2) app(z1, z2) -{ 2 }-> app(app(2, 8), app(1 + 7 + (1 + 5 + fun3), xs)) :|: z2 = 1 + (1 + 2 + 8) + xs, xs >= 0, z1 = 1 + 7 + (1 + 5 + fun3), fun3 >= 0 app(z1, z2) -{ 1 }-> app(app(2, 0), app(0, xs)) :|: fun >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + fun app(z1, z2) -{ 1 }-> app(app(2, 0), app(1 + 7 + fun, xs)) :|: fun >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + fun app(z1, z2) -{ 1 }-> app(app(2, 1 + fun + x), app(0, xs)) :|: fun >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + fun app(z1, z2) -{ 1 }-> app(app(2, 1 + fun + x), app(1 + 7 + fun, xs)) :|: fun >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + fun app(z1, z2) -{ 1 }-> app(9, app(app(0, 1), z)) :|: z >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 3 + x) + y, z2 = 1 + 9 + z app(z1, z2) -{ 1 }-> app(9, app(app(1 + 3 + 0, 1), z)) :|: z >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 3 + x) + y, z2 = 1 + 9 + z app(z1, z2) -{ 1 }-> app(0, xs) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 4) + fun) + x, xs >= 0, x >= 0, z2 = xs app(z1, z2) -{ 1 }-> app(0, app(0, xs)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, xs >= 0, x >= 0, z2 = xs app(z1, z2) -{ 1 }-> app(0, app(1 + 5 + fun, xs)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, xs >= 0, x >= 0, z2 = xs app(z1, z2) -{ 1 }-> app(1 + 5 + fun, xs) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 4) + fun) + x, xs >= 0, x >= 0, z2 = xs app(z1, z2) -{ 1 }-> app(1 + 2 + x, app(0, xs)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, xs >= 0, x >= 0, z2 = xs app(z1, z2) -{ 1 }-> app(1 + 2 + x, app(1 + 5 + fun, xs)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, xs >= 0, x >= 0, z2 = xs app(z1, z2) -{ 1 }-> 8 :|: fun >= 0, z2 = 8, z1 = 1 + 7 + fun app(z1, z2) -{ 1 }-> 8 :|: fun >= 0, z2 = 8, z1 = 1 + 5 + fun app(z1, z2) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0, z2 = v1, v1 >= 0 app(z1, z2) -{ 0 }-> 1 + x0 + x1 :|: z1 = x0, x0 >= 0, x1 >= 0, z2 = x1 encArg(z1) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = 1 + x_1 + x_2, x_2 >= 0 encArg(z1) -{ 0 }-> 9 :|: z1 = 9 encArg(z1) -{ 0 }-> 8 :|: z1 = 8 encArg(z1) -{ 0 }-> 7 :|: z1 = 7 encArg(z1) -{ 0 }-> 6 :|: z1 = 6 encArg(z1) -{ 0 }-> 5 :|: z1 = 5 encArg(z1) -{ 0 }-> 4 :|: z1 = 4 encArg(z1) -{ 0 }-> 3 :|: z1 = 3 encArg(z1) -{ 0 }-> 2 :|: z1 = 2 encArg(z1) -{ 0 }-> 10 :|: z1 = 10 encArg(z1) -{ 0 }-> 1 :|: z1 = 1 encArg(z1) -{ 0 }-> 0 :|: z1 = 0 encArg(z1) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0 encode_0 -{ 0 }-> 0 :|: encode_1 -{ 0 }-> 1 :|: encode_1 -{ 0 }-> 0 :|: encode_app(z1, z2) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = x_1, z2 = x_2, x_2 >= 0 encode_app(z1, z2) -{ 0 }-> 0 :|: z1 = v0, v0 >= 0, z2 = v1, v1 >= 0 encode_cons -{ 0 }-> 2 :|: encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 3 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 4 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 5 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 6 :|: encode_filter2 -{ 0 }-> 0 :|: encode_map -{ 0 }-> 7 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 8 :|: encode_nil -{ 0 }-> 0 :|: encode_s -{ 0 }-> 9 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 10 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(app(app(6, app(z1 - 12, x8)), z1 - 12), x8), xs2)), 1 + 5 + (z1 - 12)), 1 + (1 + 2 + x8) + xs2), xs) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x8) + xs2)) + xs, xs >= 0, xs2 >= 0, x8 >= 0, z1 - 12 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(app(3, app(9, x)), x), x)), 1 + (1 + 3 + 0) + 1), x), xs) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 5 + (1 + (1 + 3 + 0) + 1) app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(5, fun7), x)), 1 + (1 + (1 + 6 + 4) + fun7) + x7), x), xs) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x7 >= 0, x >= 0, z1 = 1 + 5 + (1 + (1 + (1 + 6 + 4) + fun7) + x7), fun7 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(2, x6), app(app(5, fun6), x))), 1 + (1 + (1 + 6 + 10) + fun6) + x6), x), xs) :|: fun6 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, x6 >= 0, z1 = 1 + 5 + (1 + (1 + (1 + 6 + 10) + fun6) + x6) app(z1, z2) -{ 2 }-> app(app(app(app(6, app(app(2, app(z1 - 14, x5)), app(app(7, z1 - 14), xs1))), 1 + 7 + (z1 - 14)), 1 + (1 + 2 + x5) + xs1), xs) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x5) + xs1)) + xs, xs >= 0, x5 >= 0, xs1 >= 0, z1 - 14 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, app(9, app(app(app(3, 0), 1), z''))), 1 + (1 + 3 + x9) + y''), 1 + 9 + z''), xs) :|: xs >= 0, z'' >= 0, z1 = 1 + 5 + (1 + (1 + 3 + x9) + y''), z2 = 1 + (1 + 2 + (1 + 9 + z'')) + xs, y'' >= 0, x9 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, 8), 1 + 7 + (z1 - 14)), 8), z2 - 12) :|: z2 - 12 >= 0, z1 - 14 >= 0 app(z1, z2) -{ 2 }-> app(app(app(app(6, 8), 1 + 5 + (z1 - 12)), 8), z2 - 12) :|: z2 - 12 >= 0, z1 - 12 >= 0 app(z1, z2) -{ 1 }-> app(app(app(app(6, 0), z1 - 6), x), xs) :|: z1 - 6 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0 app(z1, z2) -{ 1 }-> app(app(app(app(6, 1 + (z1 - 6) + x), z1 - 6), x), xs) :|: z1 - 6 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0 app(z1, z2) -{ 1 }-> app(app(app(3, 0), z2), z2) :|: z2 >= 0, z1 = 1 + (1 + 3 + 0) + 1 app(z1, z2) -{ 1 }-> app(app(app(3, 1 + 9 + z2), z2), z2) :|: z2 >= 0, z1 = 1 + (1 + 3 + 0) + 1 app(z1, z2) -{ 2 }-> app(app(2, app(app(app(app(6, app(z1 - 14, x3)), z1 - 14), x3), xs'')), app(0, xs)) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x3) + xs'')) + xs, xs >= 0, xs'' >= 0, z1 - 14 >= 0, x3 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(app(app(6, app(z1 - 14, x3)), z1 - 14), x3), xs'')), app(1 + 7 + (1 + 5 + (z1 - 14)), xs)) :|: z2 = 1 + (1 + 2 + (1 + (1 + 2 + x3) + xs'')) + xs, xs >= 0, xs'' >= 0, z1 - 14 >= 0, x3 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(app(3, app(9, x)), x), x)), app(0, xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + 3 + 0) + 1) app(z1, z2) -{ 2 }-> app(app(2, app(app(app(3, app(9, x)), x), x)), app(1 + 7 + (1 + (1 + 3 + 0) + 1), xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + 3 + 0) + 1) app(z1, z2) -{ 2 }-> app(app(2, app(app(5, fun1), x)), app(0, xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 4) + fun1) + x2), fun1 >= 0, x2 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(5, fun1), x)), app(1 + 7 + (1 + (1 + (1 + 6 + 4) + fun1) + x2), xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 4) + fun1) + x2), fun1 >= 0, x2 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, x''), app(app(5, fun''), x))), app(0, xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 10) + fun'') + x''), x >= 0, fun'' >= 0, x'' >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, x''), app(app(5, fun''), x))), app(1 + 7 + (1 + (1 + (1 + 6 + 10) + fun'') + x''), xs)) :|: xs >= 0, z2 = 1 + (1 + 2 + x) + xs, z1 = 1 + 7 + (1 + (1 + (1 + 6 + 10) + fun'') + x''), x >= 0, fun'' >= 0, x'' >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, app(z1 - 16, x')), app(app(7, z1 - 16), xs'))), app(0, xs)) :|: xs >= 0, x' >= 0, xs' >= 0, z2 = 1 + (1 + 2 + (1 + (1 + 2 + x') + xs')) + xs, z1 - 16 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(app(2, app(z1 - 16, x')), app(app(7, z1 - 16), xs'))), app(1 + 7 + (1 + 7 + (z1 - 16)), xs)) :|: xs >= 0, x' >= 0, xs' >= 0, z2 = 1 + (1 + 2 + (1 + (1 + 2 + x') + xs')) + xs, z1 - 16 >= 0 app(z1, z2) -{ 2 }-> app(app(2, app(9, app(app(app(3, 0), 1), z'))), app(0, xs)) :|: xs >= 0, x4 >= 0, z' >= 0, y' >= 0, z2 = 1 + (1 + 2 + (1 + 9 + z')) + xs, z1 = 1 + 7 + (1 + (1 + 3 + x4) + y') app(z1, z2) -{ 2 }-> app(app(2, app(9, app(app(app(3, 0), 1), z'))), app(1 + 7 + (1 + (1 + 3 + x4) + y'), xs)) :|: xs >= 0, x4 >= 0, z' >= 0, y' >= 0, z2 = 1 + (1 + 2 + (1 + 9 + z')) + xs, z1 = 1 + 7 + (1 + (1 + 3 + x4) + y') app(z1, z2) -{ 2 }-> app(app(2, 8), app(0, z2 - 12)) :|: z1 - 16 >= 0, z2 - 12 >= 0 app(z1, z2) -{ 2 }-> app(app(2, 8), app(0, z2 - 12)) :|: z2 - 12 >= 0, z1 - 14 >= 0 app(z1, z2) -{ 2 }-> app(app(2, 8), app(1 + 7 + (1 + 7 + (z1 - 16)), z2 - 12)) :|: z1 - 16 >= 0, z2 - 12 >= 0 app(z1, z2) -{ 2 }-> app(app(2, 8), app(1 + 7 + (1 + 5 + (z1 - 14)), z2 - 12)) :|: z2 - 12 >= 0, z1 - 14 >= 0 app(z1, z2) -{ 1 }-> app(app(2, 0), app(0, xs)) :|: z1 - 8 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0 app(z1, z2) -{ 1 }-> app(app(2, 0), app(1 + 7 + (z1 - 8), xs)) :|: z1 - 8 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0 app(z1, z2) -{ 1 }-> app(app(2, 1 + (z1 - 8) + x), app(0, xs)) :|: z1 - 8 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0 app(z1, z2) -{ 1 }-> app(app(2, 1 + (z1 - 8) + x), app(1 + 7 + (z1 - 8), xs)) :|: z1 - 8 >= 0, xs >= 0, z2 = 1 + (1 + 2 + x) + xs, x >= 0 app(z1, z2) -{ 1 }-> app(9, app(app(0, 1), z2 - 10)) :|: z2 - 10 >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 3 + x) + y app(z1, z2) -{ 1 }-> app(9, app(app(1 + 3 + 0, 1), z2 - 10)) :|: z2 - 10 >= 0, x >= 0, y >= 0, z1 = 1 + (1 + 3 + x) + y app(z1, z2) -{ 1 }-> app(0, z2) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 4) + fun) + x, z2 >= 0, x >= 0 app(z1, z2) -{ 1 }-> app(0, app(0, z2)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, z2 >= 0, x >= 0 app(z1, z2) -{ 1 }-> app(0, app(1 + 5 + fun, z2)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, z2 >= 0, x >= 0 app(z1, z2) -{ 1 }-> app(1 + 5 + fun, z2) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 4) + fun) + x, z2 >= 0, x >= 0 app(z1, z2) -{ 1 }-> app(1 + 2 + x, app(0, z2)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, z2 >= 0, x >= 0 app(z1, z2) -{ 1 }-> app(1 + 2 + x, app(1 + 5 + fun, z2)) :|: fun >= 0, z1 = 1 + (1 + (1 + 6 + 10) + fun) + x, z2 >= 0, x >= 0 app(z1, z2) -{ 1 }-> 8 :|: z1 - 8 >= 0, z2 = 8 app(z1, z2) -{ 1 }-> 8 :|: z1 - 6 >= 0, z2 = 8 app(z1, z2) -{ 0 }-> 0 :|: z1 >= 0, z2 >= 0 app(z1, z2) -{ 0 }-> 1 + z1 + z2 :|: z1 >= 0, z2 >= 0 encArg(z1) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z1 = 1 + x_1 + x_2, x_2 >= 0 encArg(z1) -{ 0 }-> 9 :|: z1 = 9 encArg(z1) -{ 0 }-> 8 :|: z1 = 8 encArg(z1) -{ 0 }-> 7 :|: z1 = 7 encArg(z1) -{ 0 }-> 6 :|: z1 = 6 encArg(z1) -{ 0 }-> 5 :|: z1 = 5 encArg(z1) -{ 0 }-> 4 :|: z1 = 4 encArg(z1) -{ 0 }-> 3 :|: z1 = 3 encArg(z1) -{ 0 }-> 2 :|: z1 = 2 encArg(z1) -{ 0 }-> 10 :|: z1 = 10 encArg(z1) -{ 0 }-> 1 :|: z1 = 1 encArg(z1) -{ 0 }-> 0 :|: z1 = 0 encArg(z1) -{ 0 }-> 0 :|: z1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_1 -{ 0 }-> 1 :|: encode_1 -{ 0 }-> 0 :|: encode_app(z1, z2) -{ 0 }-> app(encArg(z1), encArg(z2)) :|: z1 >= 0, z2 >= 0 encode_app(z1, z2) -{ 0 }-> 0 :|: z1 >= 0, z2 >= 0 encode_cons -{ 0 }-> 2 :|: encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 3 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 4 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 5 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 6 :|: encode_filter2 -{ 0 }-> 0 :|: encode_map -{ 0 }-> 7 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 8 :|: encode_nil -{ 0 }-> 0 :|: encode_s -{ 0 }-> 9 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 10 :|: encode_true -{ 0 }-> 0 :|: ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_f -> null_encode_f [0] encode_0 -> null_encode_0 [0] encode_1 -> null_encode_1 [0] encode_s -> null_encode_s [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_f, null_encode_0, null_encode_1, null_encode_s, null_encode_map, null_encode_nil, null_encode_cons, null_encode_filter, null_encode_filter2, null_encode_true, null_encode_false, null_app ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(map, fun), c_app(c_app(cons, x), xs)) -> app(app(cons, app(fun, x)), app(app(map, fun), xs)) [1] app(c_app(c_app(c_app(filter2, true), fun), x), xs) -> app(app(cons, x), app(app(filter, fun), xs)) [1] app(c_app(c_app(c_app(filter2, false), fun), x), xs) -> app(app(filter, fun), xs) [1] app(c_app(map, fun), nil) -> nil [1] app(c_app(filter, fun), nil) -> nil [1] app(c_app(filter, fun), c_app(c_app(cons, x), xs)) -> app(app(app(app(filter2, app(fun, x)), fun), x), xs) [1] app(c_app(c_app(f, x), y), c_app(s, z)) -> app(s, app(app(app(f, 0), 1), z)) [1] app(c_app(c_app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x) [1] encArg(f) -> f [0] encArg(0) -> 0 [0] encArg(1) -> 1 [0] encArg(s) -> s [0] encArg(map) -> map [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filter) -> filter [0] encArg(filter2) -> filter2 [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_f -> f [0] encode_0 -> 0 [0] encode_1 -> 1 [0] encode_s -> s [0] encode_map -> map [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filter -> filter [0] encode_filter2 -> filter2 [0] encode_true -> true [0] encode_false -> false [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_f -> null_encode_f [0] encode_0 -> null_encode_0 [0] encode_1 -> null_encode_1 [0] encode_s -> null_encode_s [0] encode_map -> null_encode_map [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filter -> null_encode_filter [0] encode_filter2 -> null_encode_filter2 [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app c_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app f :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app 1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app cons_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app -> map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encArg :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_0 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s: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_1 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_s :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_map :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_nil :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_cons :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_filter2 :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_true :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_encode_false :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app null_app :: map:c_app:cons:filter2:true:filter:false:nil:f:s:0:1:cons_app:null_encArg:null_encode_app:null_encode_f:null_encode_0:null_encode_1:null_encode_s:null_encode_map:null_encode_nil:null_encode_cons:null_encode_filter:null_encode_filter2:null_encode_true:null_encode_false:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: map => 7 cons => 2 filter2 => 6 true => 10 filter => 5 false => 4 nil => 8 f => 3 s => 9 0 => 0 1 => 1 null_encArg => 0 null_encode_app => 0 null_encode_f => 0 null_encode_0 => 0 null_encode_1 => 0 null_encode_s => 0 null_encode_map => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filter => 0 null_encode_filter2 => 0 null_encode_true => 0 null_encode_false => 0 null_app => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z', z'') -{ 1 }-> app(app(app(app(6, app(fun, x)), fun), x), xs) :|: fun >= 0, xs >= 0, z' = 1 + 5 + fun, z'' = 1 + (1 + 2 + x) + xs, x >= 0 app(z', z'') -{ 1 }-> app(app(app(3, app(9, x)), x), x) :|: x >= 0, z'' = x, z' = 1 + (1 + 3 + 0) + 1 app(z', z'') -{ 1 }-> app(app(5, fun), xs) :|: fun >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 6 + 4) + fun) + x app(z', z'') -{ 1 }-> app(app(2, x), app(app(5, fun), xs)) :|: fun >= 0, xs >= 0, x >= 0, z'' = xs, z' = 1 + (1 + (1 + 6 + 10) + fun) + x app(z', z'') -{ 1 }-> app(app(2, app(fun, x)), app(app(7, fun), xs)) :|: fun >= 0, xs >= 0, z' = 1 + 7 + fun, z'' = 1 + (1 + 2 + x) + xs, x >= 0 app(z', z'') -{ 1 }-> app(9, app(app(app(3, 0), 1), z)) :|: z >= 0, z' = 1 + (1 + 3 + x) + y, x >= 0, y >= 0, z'' = 1 + 9 + z app(z', z'') -{ 1 }-> 8 :|: fun >= 0, z' = 1 + 7 + fun, z'' = 8 app(z', z'') -{ 1 }-> 8 :|: fun >= 0, z' = 1 + 5 + fun, z'' = 8 app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 app(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 encArg(z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 9 :|: z' = 9 encArg(z') -{ 0 }-> 8 :|: z' = 8 encArg(z') -{ 0 }-> 7 :|: z' = 7 encArg(z') -{ 0 }-> 6 :|: z' = 6 encArg(z') -{ 0 }-> 5 :|: z' = 5 encArg(z') -{ 0 }-> 4 :|: z' = 4 encArg(z') -{ 0 }-> 3 :|: z' = 3 encArg(z') -{ 0 }-> 2 :|: z' = 2 encArg(z') -{ 0 }-> 10 :|: z' = 10 encArg(z') -{ 0 }-> 1 :|: z' = 1 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_0 -{ 0 }-> 0 :|: encode_1 -{ 0 }-> 1 :|: encode_1 -{ 0 }-> 0 :|: encode_app(z', z'') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_app(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_cons -{ 0 }-> 2 :|: encode_cons -{ 0 }-> 0 :|: encode_f -{ 0 }-> 3 :|: encode_f -{ 0 }-> 0 :|: encode_false -{ 0 }-> 4 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 5 :|: encode_filter -{ 0 }-> 0 :|: encode_filter2 -{ 0 }-> 6 :|: encode_filter2 -{ 0 }-> 0 :|: encode_map -{ 0 }-> 7 :|: encode_map -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 8 :|: encode_nil -{ 0 }-> 0 :|: encode_s -{ 0 }-> 9 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 10 :|: encode_true -{ 0 }-> 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(f) -> c ENCARG(0) -> c1 ENCARG(1) -> c2 ENCARG(s) -> c3 ENCARG(map) -> c4 ENCARG(nil) -> c5 ENCARG(cons) -> c6 ENCARG(filter) -> c7 ENCARG(filter2) -> c8 ENCARG(true) -> c9 ENCARG(false) -> c10 ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F -> c13 ENCODE_0 -> c14 ENCODE_1 -> c15 ENCODE_S -> c16 ENCODE_MAP -> c17 ENCODE_NIL -> c18 ENCODE_CONS -> c19 ENCODE_FILTER -> c20 ENCODE_FILTER2 -> c21 ENCODE_TRUE -> c22 ENCODE_FALSE -> c23 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0), APP(app(f, app(s, z0)), z0), APP(f, app(s, z0)), APP(s, z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(s, app(app(app(f, 0), 1), z2)), APP(app(app(f, 0), 1), z2), APP(app(f, 0), 1), APP(f, 0)) APP(app(map, z0), nil) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c27(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) -> c28 APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(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) -> c30(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) -> c31(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0), APP(app(f, app(s, z0)), z0), APP(f, app(s, z0)), APP(s, z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(s, app(app(app(f, 0), 1), z2)), APP(app(app(f, 0), 1), z2), APP(app(f, 0), 1), APP(f, 0)) APP(app(map, z0), nil) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c27(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) -> c28 APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(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) -> c30(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) -> c31(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_0, encode_1, encode_s, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_F, ENCODE_0, ENCODE_1, ENCODE_S, ENCODE_MAP, ENCODE_NIL, ENCODE_CONS, ENCODE_FILTER, ENCODE_FILTER2, ENCODE_TRUE, ENCODE_FALSE, APP_2 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11_3, c12_3, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24_4, c25_4, c26, c27_5, c28, c29_5, c30_4, c31_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 22 trailing nodes: ENCODE_FILTER2 -> c21 ENCODE_MAP -> c17 ENCARG(0) -> c1 ENCODE_TRUE -> c22 ENCARG(s) -> c3 ENCODE_NIL -> c18 ENCODE_0 -> c14 ENCARG(nil) -> c5 ENCODE_FILTER -> c20 ENCODE_1 -> c15 ENCARG(true) -> c9 ENCARG(filter2) -> c8 ENCARG(f) -> c ENCARG(filter) -> c7 ENCODE_FALSE -> c23 ENCARG(cons) -> c6 ENCODE_CONS -> c19 ENCARG(false) -> c10 ENCARG(map) -> c4 ENCARG(1) -> c2 ENCODE_S -> c16 ENCODE_F -> c13 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0), APP(app(f, app(s, z0)), z0), APP(f, app(s, z0)), APP(s, z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(s, app(app(app(f, 0), 1), z2)), APP(app(app(f, 0), 1), z2), APP(app(f, 0), 1), APP(f, 0)) APP(app(map, z0), nil) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c27(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) -> c28 APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(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) -> c30(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) -> c31(APP(app(filter, z0), z2), APP(filter, z0)) S tuples: APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0), APP(app(f, app(s, z0)), z0), APP(f, app(s, z0)), APP(s, z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(s, app(app(app(f, 0), 1), z2)), APP(app(app(f, 0), 1), z2), APP(app(f, 0), 1), APP(f, 0)) APP(app(map, z0), nil) -> c26 APP(app(map, z0), app(app(cons, z1), z2)) -> c27(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) -> c28 APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(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) -> c30(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) -> c31(APP(app(filter, z0), z2), APP(filter, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_0, encode_1, encode_s, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c11_3, c12_3, c24_4, c25_4, c26, c27_5, c28, c29_5, c30_4, c31_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c12(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_0, encode_1, encode_s, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c11_3, c12_3, c26, c28, c24_1, c25_1, c27_2, c29_2, c30_1, c31_1 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_0, encode_1, encode_s, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c11_3, c26, c28, c24_1, c25_1, c27_2, c29_2, c30_1, c31_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(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s encode_map -> map encode_nil -> nil encode_cons -> cons encode_filter -> filter encode_filter2 -> filter2 encode_true -> true encode_false -> false app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_f, encode_0, encode_1, encode_s, encode_map, encode_nil, encode_cons, encode_filter, encode_filter2, encode_true, encode_false Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c11_3, c26, c28, c24_1, c25_1, c27_2, c29_2, c30_1, c31_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_f -> f encode_0 -> 0 encode_1 -> 1 encode_s -> s 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(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c11_3, c26, c28, c24_1, c25_1, c27_2, c29_2, c30_1, c31_1, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0), ENCARG(1)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(f, x1)) -> c11(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(APP(1, encArg(x1)), ENCARG(1), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0), ENCARG(f)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0), ENCARG(1)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0), ENCARG(map)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0), ENCARG(filter2)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(f, x1)) -> c11(APP(f, encArg(x1)), ENCARG(f), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(APP(1, encArg(x1)), ENCARG(1), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(APP(map, encArg(x1)), ENCARG(map), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(APP(filter2, encArg(x1)), ENCARG(filter2), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(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, c28, c24_1, c25_1, c27_2, c29_2, c30_1, c31_1, c_1, c11_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 33 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(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, c28, c24_1, c25_1, c27_2, c29_2, c30_1, c31_1, c_1, c11_3, c11_2, c11_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c29(APP(app(app(app(filter2, app(z0, z1)), z0), z1), z2), APP(z0, z1)) by APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(app(filter2, app(s, app(app(app(f, 0), 1), z2))), app(app(f, z0), z1)), app(s, z2)), x2), APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(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)) -> c29(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)) -> c29(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)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c29(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)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(app(filter2, app(s, app(app(app(f, 0), 1), z2))), app(app(f, z0), z1)), app(s, z2)), x2), APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(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)) -> c29(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)) -> c29(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)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c29(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)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(app(filter2, app(s, app(app(app(f, 0), 1), z2))), app(app(f, z0), z1)), app(s, z2)), x2), APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(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)) -> c29(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)) -> c29(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)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(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)) -> c29(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)) -> c29(APP(app(app(app(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c_1, c11_3, c11_2, c11_1, c29_2 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c_1, c11_3, c11_2, c11_1, c29_2, c29_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) 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(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(1, x1) -> c(APP(1, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) 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(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(1, x1) -> c(APP(1, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_3, c11_2, c11_1, c29_2, c29_1, c_1 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCODE_APP(map, x1) -> c(APP(map, encArg(x1))) ENCODE_APP(f, x1) -> c(APP(f, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(1, x1) -> c(APP(1, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(filter2, x1) -> c(APP(filter2, encArg(x1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_3, c11_2, c11_1, c29_2, c29_1, c_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c11(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(APP(1, app(encArg(x1), encArg(x2))), ENCARG(1), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(APP(f, app(encArg(x1), encArg(x2))), ENCARG(f), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(APP(1, app(encArg(x1), encArg(x2))), ENCARG(1), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(APP(map, app(encArg(x1), encArg(x2))), ENCARG(map), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(APP(filter2, app(encArg(x1), encArg(x2))), ENCARG(filter2), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_3, c11_2, c11_1, c29_2, c29_1, c_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 22 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_3, c11_2, c11_1, c29_2, c29_1, c_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c11(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1)), ENCARG(1)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c11(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1))), ENCARG(cons_app(x0, x1)), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(APP(app(1, encArg(x1)), encArg(x2)), ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1)), ENCARG(f)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1)), ENCARG(1)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1)), ENCARG(map)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1)), ENCARG(filter2)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(APP(app(f, encArg(x1)), encArg(x2)), ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(APP(app(1, encArg(x1)), encArg(x2)), ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(APP(app(filter2, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 20 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, f)) -> c11(APP(encArg(x0), f), ENCARG(x0)) by ENCARG(cons_app(f, f)) -> c11(APP(f, f), ENCARG(f)) ENCARG(cons_app(0, f)) -> c11(APP(0, f), ENCARG(0)) ENCARG(cons_app(1, f)) -> c11(APP(1, f), ENCARG(1)) ENCARG(cons_app(s, f)) -> c11(APP(s, f), ENCARG(s)) ENCARG(cons_app(map, f)) -> c11(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c11(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c11(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c11(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c11(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c11(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c11(APP(false, f), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), f)) -> c11(APP(app(encArg(z0), encArg(z1)), f), ENCARG(cons_app(z0, z1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, f)) -> c11(APP(f, f), ENCARG(f)) ENCARG(cons_app(0, f)) -> c11(APP(0, f), ENCARG(0)) ENCARG(cons_app(1, f)) -> c11(APP(1, f), ENCARG(1)) ENCARG(cons_app(s, f)) -> c11(APP(s, f), ENCARG(s)) ENCARG(cons_app(map, f)) -> c11(APP(map, f), ENCARG(map)) ENCARG(cons_app(nil, f)) -> c11(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(cons, f)) -> c11(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(filter, f)) -> c11(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(filter2, f)) -> c11(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(true, f)) -> c11(APP(true, f), ENCARG(true)) ENCARG(cons_app(false, f)) -> c11(APP(false, f), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(filter2, f)) -> c11(APP(filter2, f), ENCARG(filter2)) ENCARG(cons_app(cons, f)) -> c11(APP(cons, f), ENCARG(cons)) ENCARG(cons_app(map, f)) -> c11(APP(map, f), ENCARG(map)) ENCARG(cons_app(1, f)) -> c11(APP(1, f), ENCARG(1)) ENCARG(cons_app(nil, f)) -> c11(APP(nil, f), ENCARG(nil)) ENCARG(cons_app(0, f)) -> c11(APP(0, f), ENCARG(0)) ENCARG(cons_app(false, f)) -> c11(APP(false, f), ENCARG(false)) ENCARG(cons_app(true, f)) -> c11(APP(true, f), ENCARG(true)) ENCARG(cons_app(f, f)) -> c11(APP(f, f), ENCARG(f)) ENCARG(cons_app(filter, f)) -> c11(APP(filter, f), ENCARG(filter)) ENCARG(cons_app(s, f)) -> c11(APP(s, f), ENCARG(s)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, 0)) -> c11(APP(encArg(x0), 0), ENCARG(x0)) by ENCARG(cons_app(f, 0)) -> c11(APP(f, 0), ENCARG(f)) ENCARG(cons_app(0, 0)) -> c11(APP(0, 0), ENCARG(0)) ENCARG(cons_app(1, 0)) -> c11(APP(1, 0), ENCARG(1)) ENCARG(cons_app(s, 0)) -> c11(APP(s, 0), ENCARG(s)) ENCARG(cons_app(map, 0)) -> c11(APP(map, 0), ENCARG(map)) ENCARG(cons_app(nil, 0)) -> c11(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c11(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filter, 0)) -> c11(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(filter2, 0)) -> c11(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(true, 0)) -> c11(APP(true, 0), ENCARG(true)) ENCARG(cons_app(false, 0)) -> c11(APP(false, 0), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), 0)) -> c11(APP(app(encArg(z0), encArg(z1)), 0), ENCARG(cons_app(z0, z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 0)) -> c11(APP(f, 0), ENCARG(f)) ENCARG(cons_app(0, 0)) -> c11(APP(0, 0), ENCARG(0)) ENCARG(cons_app(1, 0)) -> c11(APP(1, 0), ENCARG(1)) ENCARG(cons_app(s, 0)) -> c11(APP(s, 0), ENCARG(s)) ENCARG(cons_app(map, 0)) -> c11(APP(map, 0), ENCARG(map)) ENCARG(cons_app(nil, 0)) -> c11(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c11(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filter, 0)) -> c11(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(filter2, 0)) -> c11(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(true, 0)) -> c11(APP(true, 0), ENCARG(true)) ENCARG(cons_app(false, 0)) -> c11(APP(false, 0), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(map, 0)) -> c11(APP(map, 0), ENCARG(map)) ENCARG(cons_app(true, 0)) -> c11(APP(true, 0), ENCARG(true)) ENCARG(cons_app(filter, 0)) -> c11(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(s, 0)) -> c11(APP(s, 0), ENCARG(s)) ENCARG(cons_app(filter2, 0)) -> c11(APP(filter2, 0), ENCARG(filter2)) ENCARG(cons_app(0, 0)) -> c11(APP(0, 0), ENCARG(0)) ENCARG(cons_app(false, 0)) -> c11(APP(false, 0), ENCARG(false)) ENCARG(cons_app(f, 0)) -> c11(APP(f, 0), ENCARG(f)) ENCARG(cons_app(cons, 0)) -> c11(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(nil, 0)) -> c11(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(1, 0)) -> c11(APP(1, 0), ENCARG(1)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, 1)) -> c11(APP(encArg(x0), 1), ENCARG(x0)) by ENCARG(cons_app(f, 1)) -> c11(APP(f, 1), ENCARG(f)) ENCARG(cons_app(0, 1)) -> c11(APP(0, 1), ENCARG(0)) ENCARG(cons_app(1, 1)) -> c11(APP(1, 1), ENCARG(1)) ENCARG(cons_app(s, 1)) -> c11(APP(s, 1), ENCARG(s)) ENCARG(cons_app(map, 1)) -> c11(APP(map, 1), ENCARG(map)) ENCARG(cons_app(nil, 1)) -> c11(APP(nil, 1), ENCARG(nil)) ENCARG(cons_app(cons, 1)) -> c11(APP(cons, 1), ENCARG(cons)) ENCARG(cons_app(filter, 1)) -> c11(APP(filter, 1), ENCARG(filter)) ENCARG(cons_app(filter2, 1)) -> c11(APP(filter2, 1), ENCARG(filter2)) ENCARG(cons_app(true, 1)) -> c11(APP(true, 1), ENCARG(true)) ENCARG(cons_app(false, 1)) -> c11(APP(false, 1), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), 1)) -> c11(APP(app(encArg(z0), encArg(z1)), 1), ENCARG(cons_app(z0, z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, 1)) -> c11(APP(f, 1), ENCARG(f)) ENCARG(cons_app(0, 1)) -> c11(APP(0, 1), ENCARG(0)) ENCARG(cons_app(1, 1)) -> c11(APP(1, 1), ENCARG(1)) ENCARG(cons_app(s, 1)) -> c11(APP(s, 1), ENCARG(s)) ENCARG(cons_app(map, 1)) -> c11(APP(map, 1), ENCARG(map)) ENCARG(cons_app(nil, 1)) -> c11(APP(nil, 1), ENCARG(nil)) ENCARG(cons_app(cons, 1)) -> c11(APP(cons, 1), ENCARG(cons)) ENCARG(cons_app(filter, 1)) -> c11(APP(filter, 1), ENCARG(filter)) ENCARG(cons_app(filter2, 1)) -> c11(APP(filter2, 1), ENCARG(filter2)) ENCARG(cons_app(true, 1)) -> c11(APP(true, 1), ENCARG(true)) ENCARG(cons_app(false, 1)) -> c11(APP(false, 1), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(false, 1)) -> c11(APP(false, 1), ENCARG(false)) ENCARG(cons_app(f, 1)) -> c11(APP(f, 1), ENCARG(f)) ENCARG(cons_app(true, 1)) -> c11(APP(true, 1), ENCARG(true)) ENCARG(cons_app(s, 1)) -> c11(APP(s, 1), ENCARG(s)) ENCARG(cons_app(0, 1)) -> c11(APP(0, 1), ENCARG(0)) ENCARG(cons_app(filter2, 1)) -> c11(APP(filter2, 1), ENCARG(filter2)) ENCARG(cons_app(filter, 1)) -> c11(APP(filter, 1), ENCARG(filter)) ENCARG(cons_app(cons, 1)) -> c11(APP(cons, 1), ENCARG(cons)) ENCARG(cons_app(1, 1)) -> c11(APP(1, 1), ENCARG(1)) ENCARG(cons_app(nil, 1)) -> c11(APP(nil, 1), ENCARG(nil)) ENCARG(cons_app(map, 1)) -> c11(APP(map, 1), ENCARG(map)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, s)) -> c11(APP(encArg(x0), s), ENCARG(x0)) by ENCARG(cons_app(f, s)) -> c11(APP(f, s), ENCARG(f)) ENCARG(cons_app(0, s)) -> c11(APP(0, s), ENCARG(0)) ENCARG(cons_app(1, s)) -> c11(APP(1, s), ENCARG(1)) ENCARG(cons_app(s, s)) -> c11(APP(s, s), ENCARG(s)) ENCARG(cons_app(map, s)) -> c11(APP(map, s), ENCARG(map)) ENCARG(cons_app(nil, s)) -> c11(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(cons, s)) -> c11(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(filter, s)) -> c11(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(filter2, s)) -> c11(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c11(APP(true, s), ENCARG(true)) ENCARG(cons_app(false, s)) -> c11(APP(false, s), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), s)) -> c11(APP(app(encArg(z0), encArg(z1)), s), ENCARG(cons_app(z0, z1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, s)) -> c11(APP(f, s), ENCARG(f)) ENCARG(cons_app(0, s)) -> c11(APP(0, s), ENCARG(0)) ENCARG(cons_app(1, s)) -> c11(APP(1, s), ENCARG(1)) ENCARG(cons_app(s, s)) -> c11(APP(s, s), ENCARG(s)) ENCARG(cons_app(map, s)) -> c11(APP(map, s), ENCARG(map)) ENCARG(cons_app(nil, s)) -> c11(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(cons, s)) -> c11(APP(cons, s), ENCARG(cons)) ENCARG(cons_app(filter, s)) -> c11(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(filter2, s)) -> c11(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c11(APP(true, s), ENCARG(true)) ENCARG(cons_app(false, s)) -> c11(APP(false, s), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(nil, s)) -> c11(APP(nil, s), ENCARG(nil)) ENCARG(cons_app(0, s)) -> c11(APP(0, s), ENCARG(0)) ENCARG(cons_app(1, s)) -> c11(APP(1, s), ENCARG(1)) ENCARG(cons_app(filter, s)) -> c11(APP(filter, s), ENCARG(filter)) ENCARG(cons_app(s, s)) -> c11(APP(s, s), ENCARG(s)) ENCARG(cons_app(false, s)) -> c11(APP(false, s), ENCARG(false)) ENCARG(cons_app(filter2, s)) -> c11(APP(filter2, s), ENCARG(filter2)) ENCARG(cons_app(true, s)) -> c11(APP(true, s), ENCARG(true)) ENCARG(cons_app(f, s)) -> c11(APP(f, s), ENCARG(f)) ENCARG(cons_app(map, s)) -> c11(APP(map, s), ENCARG(map)) ENCARG(cons_app(cons, s)) -> c11(APP(cons, s), ENCARG(cons)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, map)) -> c11(APP(encArg(x0), map), ENCARG(x0)) by ENCARG(cons_app(f, map)) -> c11(APP(f, map), ENCARG(f)) ENCARG(cons_app(0, map)) -> c11(APP(0, map), ENCARG(0)) ENCARG(cons_app(1, map)) -> c11(APP(1, map), ENCARG(1)) ENCARG(cons_app(s, map)) -> c11(APP(s, map), ENCARG(s)) ENCARG(cons_app(map, map)) -> c11(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c11(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c11(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c11(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c11(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c11(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c11(APP(false, map), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), map)) -> c11(APP(app(encArg(z0), encArg(z1)), map), ENCARG(cons_app(z0, z1))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, map)) -> c11(APP(f, map), ENCARG(f)) ENCARG(cons_app(0, map)) -> c11(APP(0, map), ENCARG(0)) ENCARG(cons_app(1, map)) -> c11(APP(1, map), ENCARG(1)) ENCARG(cons_app(s, map)) -> c11(APP(s, map), ENCARG(s)) ENCARG(cons_app(map, map)) -> c11(APP(map, map), ENCARG(map)) ENCARG(cons_app(nil, map)) -> c11(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(cons, map)) -> c11(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(filter, map)) -> c11(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(filter2, map)) -> c11(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(true, map)) -> c11(APP(true, map), ENCARG(true)) ENCARG(cons_app(false, map)) -> c11(APP(false, map), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(filter2, map)) -> c11(APP(filter2, map), ENCARG(filter2)) ENCARG(cons_app(filter, map)) -> c11(APP(filter, map), ENCARG(filter)) ENCARG(cons_app(cons, map)) -> c11(APP(cons, map), ENCARG(cons)) ENCARG(cons_app(f, map)) -> c11(APP(f, map), ENCARG(f)) ENCARG(cons_app(nil, map)) -> c11(APP(nil, map), ENCARG(nil)) ENCARG(cons_app(false, map)) -> c11(APP(false, map), ENCARG(false)) ENCARG(cons_app(map, map)) -> c11(APP(map, map), ENCARG(map)) ENCARG(cons_app(true, map)) -> c11(APP(true, map), ENCARG(true)) ENCARG(cons_app(s, map)) -> c11(APP(s, map), ENCARG(s)) ENCARG(cons_app(0, map)) -> c11(APP(0, map), ENCARG(0)) ENCARG(cons_app(1, map)) -> c11(APP(1, map), ENCARG(1)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c11(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(f, nil)) -> c11(APP(f, nil), ENCARG(f)) ENCARG(cons_app(0, nil)) -> c11(APP(0, nil), ENCARG(0)) ENCARG(cons_app(1, nil)) -> c11(APP(1, nil), ENCARG(1)) ENCARG(cons_app(s, nil)) -> c11(APP(s, nil), ENCARG(s)) ENCARG(cons_app(map, nil)) -> c11(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c11(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c11(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c11(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c11(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c11(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c11(APP(false, nil), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c11(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, nil)) -> c11(APP(f, nil), ENCARG(f)) ENCARG(cons_app(0, nil)) -> c11(APP(0, nil), ENCARG(0)) ENCARG(cons_app(1, nil)) -> c11(APP(1, nil), ENCARG(1)) ENCARG(cons_app(s, nil)) -> c11(APP(s, nil), ENCARG(s)) ENCARG(cons_app(map, nil)) -> c11(APP(map, nil), ENCARG(map)) ENCARG(cons_app(nil, nil)) -> c11(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c11(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filter, nil)) -> c11(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(filter2, nil)) -> c11(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(true, nil)) -> c11(APP(true, nil), ENCARG(true)) ENCARG(cons_app(false, nil)) -> c11(APP(false, nil), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(nil, nil)) -> c11(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(true, nil)) -> c11(APP(true, nil), ENCARG(true)) ENCARG(cons_app(cons, nil)) -> c11(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(false, nil)) -> c11(APP(false, nil), ENCARG(false)) ENCARG(cons_app(filter2, nil)) -> c11(APP(filter2, nil), ENCARG(filter2)) ENCARG(cons_app(f, nil)) -> c11(APP(f, nil), ENCARG(f)) ENCARG(cons_app(0, nil)) -> c11(APP(0, nil), ENCARG(0)) ENCARG(cons_app(filter, nil)) -> c11(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(map, nil)) -> c11(APP(map, nil), ENCARG(map)) ENCARG(cons_app(s, nil)) -> c11(APP(s, nil), ENCARG(s)) ENCARG(cons_app(1, nil)) -> c11(APP(1, nil), ENCARG(1)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c11(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(f, cons)) -> c11(APP(f, cons), ENCARG(f)) ENCARG(cons_app(0, cons)) -> c11(APP(0, cons), ENCARG(0)) ENCARG(cons_app(1, cons)) -> c11(APP(1, cons), ENCARG(1)) ENCARG(cons_app(s, cons)) -> c11(APP(s, cons), ENCARG(s)) ENCARG(cons_app(map, cons)) -> c11(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c11(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c11(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c11(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c11(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c11(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c11(APP(false, cons), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c11(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, cons)) -> c11(APP(f, cons), ENCARG(f)) ENCARG(cons_app(0, cons)) -> c11(APP(0, cons), ENCARG(0)) ENCARG(cons_app(1, cons)) -> c11(APP(1, cons), ENCARG(1)) ENCARG(cons_app(s, cons)) -> c11(APP(s, cons), ENCARG(s)) ENCARG(cons_app(map, cons)) -> c11(APP(map, cons), ENCARG(map)) ENCARG(cons_app(nil, cons)) -> c11(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c11(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(filter, cons)) -> c11(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(filter2, cons)) -> c11(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(true, cons)) -> c11(APP(true, cons), ENCARG(true)) ENCARG(cons_app(false, cons)) -> c11(APP(false, cons), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(filter2, cons)) -> c11(APP(filter2, cons), ENCARG(filter2)) ENCARG(cons_app(nil, cons)) -> c11(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(1, cons)) -> c11(APP(1, cons), ENCARG(1)) ENCARG(cons_app(0, cons)) -> c11(APP(0, cons), ENCARG(0)) ENCARG(cons_app(map, cons)) -> c11(APP(map, cons), ENCARG(map)) ENCARG(cons_app(true, cons)) -> c11(APP(true, cons), ENCARG(true)) ENCARG(cons_app(cons, cons)) -> c11(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(false, cons)) -> c11(APP(false, cons), ENCARG(false)) ENCARG(cons_app(filter, cons)) -> c11(APP(filter, cons), ENCARG(filter)) ENCARG(cons_app(s, cons)) -> c11(APP(s, cons), ENCARG(s)) ENCARG(cons_app(f, cons)) -> c11(APP(f, cons), ENCARG(f)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, filter)) -> c11(APP(encArg(x0), filter), ENCARG(x0)) by ENCARG(cons_app(f, filter)) -> c11(APP(f, filter), ENCARG(f)) ENCARG(cons_app(0, filter)) -> c11(APP(0, filter), ENCARG(0)) ENCARG(cons_app(1, filter)) -> c11(APP(1, filter), ENCARG(1)) ENCARG(cons_app(s, filter)) -> c11(APP(s, filter), ENCARG(s)) ENCARG(cons_app(map, filter)) -> c11(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c11(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c11(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c11(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c11(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c11(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c11(APP(false, filter), ENCARG(false)) ENCARG(cons_app(cons_app(z0, z1), filter)) -> c11(APP(app(encArg(z0), encArg(z1)), filter), ENCARG(cons_app(z0, z1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(f, filter)) -> c11(APP(f, filter), ENCARG(f)) ENCARG(cons_app(0, filter)) -> c11(APP(0, filter), ENCARG(0)) ENCARG(cons_app(1, filter)) -> c11(APP(1, filter), ENCARG(1)) ENCARG(cons_app(s, filter)) -> c11(APP(s, filter), ENCARG(s)) ENCARG(cons_app(map, filter)) -> c11(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c11(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c11(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(filter, filter)) -> c11(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(filter2, filter)) -> c11(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(true, filter)) -> c11(APP(true, filter), ENCARG(true)) ENCARG(cons_app(false, filter)) -> c11(APP(false, filter), ENCARG(false)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: ENCARG(cons_app(true, filter)) -> c11(APP(true, filter), ENCARG(true)) ENCARG(cons_app(0, filter)) -> c11(APP(0, filter), ENCARG(0)) ENCARG(cons_app(f, filter)) -> c11(APP(f, filter), ENCARG(f)) ENCARG(cons_app(filter2, filter)) -> c11(APP(filter2, filter), ENCARG(filter2)) ENCARG(cons_app(filter, filter)) -> c11(APP(filter, filter), ENCARG(filter)) ENCARG(cons_app(s, filter)) -> c11(APP(s, filter), ENCARG(s)) ENCARG(cons_app(false, filter)) -> c11(APP(false, filter), ENCARG(false)) ENCARG(cons_app(map, filter)) -> c11(APP(map, filter), ENCARG(map)) ENCARG(cons_app(nil, filter)) -> c11(APP(nil, filter), ENCARG(nil)) ENCARG(cons_app(cons, filter)) -> c11(APP(cons, filter), ENCARG(cons)) ENCARG(cons_app(1, filter)) -> c11(APP(1, filter), ENCARG(1)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(f) -> f encArg(0) -> 0 encArg(1) -> 1 encArg(s) -> s encArg(map) -> map encArg(nil) -> nil encArg(cons) -> cons encArg(filter) -> filter encArg(filter2) -> filter2 encArg(true) -> true encArg(false) -> false encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(app(f, 0), 1), z0) -> app(app(app(f, app(s, z0)), z0), z0) app(app(app(f, z0), z1), app(s, z2)) -> app(s, app(app(app(f, 0), 1), z2)) app(app(map, z0), nil) -> nil app(app(map, z0), app(app(cons, z1), z2)) -> app(app(cons, app(z0, z1)), app(app(map, z0), z2)) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(app(filter2, app(z0, z1)), z0), z1), z2) app(app(app(app(filter2, true), z0), z1), z2) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(app(filter2, false), z0), z1), z2) -> app(app(filter, z0), z2) Tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, filter2)) -> c11(APP(encArg(x0), filter2), ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c11(APP(encArg(x0), true), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c11(APP(encArg(x0), false), ENCARG(x0)) ENCARG(cons_app(f, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(1, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(map, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(filter2, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c11(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c11(ENCARG(x1)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, true), z0), z1), z2)) ENCODE_APP(x0, f) -> c(APP(encArg(x0), f)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, 1) -> c(APP(encArg(x0), 1)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, map) -> c(APP(encArg(x0), map)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, filter2) -> c(APP(encArg(x0), filter2)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, f))) -> c11(APP(encArg(x0), app(encArg(x1), f)), ENCARG(x0), ENCARG(cons_app(x1, f))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c11(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, 1))) -> c11(APP(encArg(x0), app(encArg(x1), 1)), ENCARG(x0), ENCARG(cons_app(x1, 1))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c11(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, map))) -> c11(APP(encArg(x0), app(encArg(x1), map)), ENCARG(x0), ENCARG(cons_app(x1, map))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c11(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c11(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c11(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, filter2))) -> c11(APP(encArg(x0), app(encArg(x1), filter2)), ENCARG(x0), ENCARG(cons_app(x1, filter2))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c11(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c11(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c11(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(f, x2))) -> c11(APP(encArg(x0), app(f, encArg(x2))), ENCARG(x0), ENCARG(cons_app(f, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c11(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(1, x2))) -> c11(APP(encArg(x0), app(1, encArg(x2))), ENCARG(x0), ENCARG(cons_app(1, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c11(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(map, x2))) -> c11(APP(encArg(x0), app(map, encArg(x2))), ENCARG(x0), ENCARG(cons_app(map, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c11(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c11(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c11(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(filter2, x2))) -> c11(APP(encArg(x0), app(filter2, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter2, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c11(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c11(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c11(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c11(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(f, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(1, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(map, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter2, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c11(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, f), x2)) -> c11(APP(app(encArg(x0), f), encArg(x2)), ENCARG(cons_app(x0, f)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c11(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 1), x2)) -> c11(APP(app(encArg(x0), 1), encArg(x2)), ENCARG(cons_app(x0, 1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c11(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, map), x2)) -> c11(APP(app(encArg(x0), map), encArg(x2)), ENCARG(cons_app(x0, map)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c11(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c11(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c11(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter2), x2)) -> c11(APP(app(encArg(x0), filter2), encArg(x2)), ENCARG(cons_app(x0, filter2)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c11(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c11(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c11(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(map, x1), x2)) -> c11(APP(app(map, encArg(x1)), encArg(x2)), ENCARG(cons_app(map, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c11(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c11(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), f)) -> c11(APP(app(encArg(x0), encArg(x1)), f), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c11(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 1)) -> c11(APP(app(encArg(x0), encArg(x1)), 1), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c11(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), map)) -> c11(APP(app(encArg(x0), encArg(x1)), map), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c11(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c11(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c11(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter2)) -> c11(APP(app(encArg(x0), encArg(x1)), filter2), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c11(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c11(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(f, x1), x2)) -> c11(ENCARG(cons_app(f, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c11(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(1, x1), x2)) -> c11(ENCARG(cons_app(1, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c11(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c11(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c11(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter2, x1), x2)) -> c11(ENCARG(cons_app(filter2, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c11(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c11(ENCARG(cons_app(false, x1)), ENCARG(x2)) S tuples: APP(app(map, z0), nil) -> c26 APP(app(filter, z0), nil) -> c28 APP(app(app(f, 0), 1), z0) -> c24(APP(app(app(f, app(s, z0)), z0), z0)) APP(app(app(f, z0), z1), app(s, z2)) -> c25(APP(app(app(f, 0), 1), z2)) APP(app(map, z0), app(app(cons, z1), z2)) -> c27(APP(z0, z1), APP(app(map, z0), z2)) APP(app(app(app(filter2, true), z0), z1), z2) -> c30(APP(app(filter, z0), z2)) APP(app(app(app(filter2, false), z0), z1), z2) -> c31(APP(app(filter, z0), z2)) APP(app(filter, app(app(f, 0), 1)), app(app(cons, z0), x2)) -> c29(APP(app(app(app(filter2, app(app(app(f, app(s, z0)), z0), z0)), app(app(f, 0), 1)), z0), x2), APP(app(app(f, 0), 1), z0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(app(app(filter2, app(app(app(app(filter2, app(z0, z1)), z0), z1), z2)), app(filter, z0)), app(app(cons, z1), z2)), x2), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(app(filter2, false), z0), z1)), app(app(cons, z2), x2)) -> c29(APP(app(app(app(filter2, app(app(filter, z0), z2)), app(app(app(filter2, false), z0), z1)), z2), x2), APP(app(app(app(filter2, false), z0), z1), z2)) APP(app(filter, app(app(f, z0), z1)), app(app(cons, app(s, z2)), x2)) -> c29(APP(app(app(f, z0), z1), app(s, z2))) APP(app(filter, app(map, z0)), app(app(cons, nil), x2)) -> c29(APP(app(map, z0), nil)) APP(app(filter, app(map, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c29(APP(app(map, z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c29(APP(app(filter, z0), nil)) APP(app(filter, app(app(app(filter2, true), z0), z1)), app(app(cons, z2), x2)) -> c29(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, c28, c24_1, c25_1, c27_2, c30_1, c31_1, c11_2, c11_1, c29_2, c29_1, c_1, c11_3