/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 215 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), 5801 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), 2 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 13 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 11 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 96 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 67 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 46 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 4 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 59 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms] (134) 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(neq, 0), 0) -> false app(app(neq, 0), app(s, y)) -> true app(app(neq, app(s, x)), 0) -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0)) 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(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero ---------------------------------------- (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(neq, 0), 0) -> false app(app(neq, 0), app(s, y)) -> true app(app(neq, app(s, x)), 0) -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0)) The (relative) TRS S consists of the following rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero 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(neq, 0), 0) -> false app(app(neq, 0), app(s, y)) -> true app(app(neq, app(s, x)), 0) -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0)) The (relative) TRS S consists of the following rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero 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(neq, 0), 0) -> false app(app(neq, 0), app(s, y)) -> true app(app(neq, app(s, x)), 0) -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0)) The (relative) TRS S consists of the following rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero 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(neq, 0'), 0') -> false app(app(neq, 0'), app(s, y)) -> true app(app(neq, app(s, x)), 0') -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0')) The (relative) TRS S consists of the following rules: encArg(neq) -> neq encArg(0') -> 0' encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0' encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(app(neq, 0'), 0') -> false app(app(neq, 0'), app(s, y)) -> true app(app(neq, app(s, x)), 0') -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0')) encArg(neq) -> neq encArg(0') -> 0' encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0' encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero Types: app :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero neq :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero 0' :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero false :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero s :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero true :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero filter :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero nil :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero cons :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero filtersub :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero nonzero :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encArg :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero cons_app :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero cons_nonzero :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_app :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_neq :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_0 :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_false :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_s :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_true :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_filter :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_nil :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_cons :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_filtersub :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_nonzero :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero hole_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero1_3 :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3 :: Nat -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero ---------------------------------------- (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(neq, 0'), 0') -> false app(app(neq, 0'), app(s, y)) -> true app(app(neq, app(s, x)), 0') -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0')) encArg(neq) -> neq encArg(0') -> 0' encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0' encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero Types: app :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero neq :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero 0' :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero false :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero s :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero true :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero filter :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero nil :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero cons :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero filtersub :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero nonzero :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encArg :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero cons_app :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero cons_nonzero :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_app :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_neq :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_0 :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_false :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_s :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_true :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_filter :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_nil :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_cons :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_filtersub :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero encode_nonzero :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero hole_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero1_3 :: neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3 :: Nat -> neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero Generator Equations: gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(0) <=> neq gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(+(x, 1)) <=> cons_app(neq, gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_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_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(n2249_3)) -> *3_3, rt in Omega(0) Induction Base: encArg(gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(0)) Induction Step: encArg(gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(+(n2249_3, 1))) ->_R^Omega(0) app(encArg(neq), encArg(gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(n2249_3))) ->_R^Omega(0) app(neq, encArg(gen_neq:0':false:s:true:filter:nil:cons:filtersub:cons_app:cons_nonzero2_3(n2249_3))) ->_IH app(neq, *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(neq, 0), 0) -> false app(app(neq, 0), app(s, y)) -> true app(app(neq, app(s, x)), 0) -> true app(app(neq, app(s, x)), app(s, y)) -> app(app(neq, x), y) app(app(filter, f), nil) -> nil app(app(filter, f), app(app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(app(app(filtersub, true), f), app(app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(app(app(filtersub, false), f), app(app(cons, y), ys)) -> app(app(filter, f), ys) nonzero -> app(filter, app(neq, 0)) encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero 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: nonzero -> app(filter, app(neq, 0)) app(c_app(neq, c_app(s, x)), 0) -> true app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(app(filter, f), ys) app(c_app(neq, 0), 0) -> false app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(app(neq, x), y) app(c_app(filter, f), nil) -> nil app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) app(c_app(neq, 0), c_app(s, y)) -> true The (relative) TRS S consists of the following rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_nonzero) -> nonzero encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero 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: nonzero -> app(filter, app(neq, 0)) [1] app(c_app(neq, c_app(s, x)), 0) -> true [1] app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(app(filter, f), ys) [1] app(c_app(neq, 0), 0) -> false [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) [1] app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(app(neq, x), y) [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) [1] app(c_app(neq, 0), c_app(s, y)) -> true [1] encArg(neq) -> neq [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s) -> s [0] encArg(true) -> true [0] encArg(filter) -> filter [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filtersub) -> filtersub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encArg(cons_nonzero) -> nonzero [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_neq -> neq [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s -> s [0] encode_true -> true [0] encode_filter -> filter [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filtersub -> filtersub [0] encode_nonzero -> nonzero [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: nonzero -> app(filter, app(neq, 0)) [1] app(c_app(neq, c_app(s, x)), 0) -> true [1] app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(app(filter, f), ys) [1] app(c_app(neq, 0), 0) -> false [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) [1] app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(app(neq, x), y) [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) [1] app(c_app(neq, 0), c_app(s, y)) -> true [1] encArg(neq) -> neq [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s) -> s [0] encArg(true) -> true [0] encArg(filter) -> filter [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filtersub) -> filtersub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encArg(cons_nonzero) -> nonzero [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_neq -> neq [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s -> s [0] encode_true -> true [0] encode_filter -> filter [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filtersub -> filtersub [0] encode_nonzero -> nonzero [0] app(x0, x1) -> c_app(x0, x1) [0] The TRS has the following type information: nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero 0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero c_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero cons_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero cons_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero 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: nonzero encArg_1 encode_app_2 encode_neq encode_0 encode_false encode_s encode_true encode_filter encode_nil encode_cons encode_filtersub encode_nonzero app_2 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_neq -> null_encode_neq [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s -> null_encode_s [0] encode_true -> null_encode_true [0] encode_filter -> null_encode_filter [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filtersub -> null_encode_filtersub [0] encode_nonzero -> null_encode_nonzero [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_neq, null_encode_0, null_encode_false, null_encode_s, null_encode_true, null_encode_filter, null_encode_nil, null_encode_cons, null_encode_filtersub, null_encode_nonzero, 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: nonzero -> app(filter, app(neq, 0)) [1] app(c_app(neq, c_app(s, x)), 0) -> true [1] app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(app(filter, f), ys) [1] app(c_app(neq, 0), 0) -> false [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) [1] app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(app(neq, x), y) [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) [1] app(c_app(neq, 0), c_app(s, y)) -> true [1] encArg(neq) -> neq [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s) -> s [0] encArg(true) -> true [0] encArg(filter) -> filter [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filtersub) -> filtersub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encArg(cons_nonzero) -> nonzero [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_neq -> neq [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s -> s [0] encode_true -> true [0] encode_filter -> filter [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filtersub -> filtersub [0] encode_nonzero -> nonzero [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_neq -> null_encode_neq [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s -> null_encode_s [0] encode_true -> null_encode_true [0] encode_filter -> null_encode_filter [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filtersub -> null_encode_filtersub [0] encode_nonzero -> null_encode_nonzero [0] app(v0, v1) -> null_app [0] The TRS has the following type information: nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app 0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app c_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero: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: nonzero -> app(filter, c_app(neq, 0)) [1] nonzero -> app(filter, null_app) [1] app(c_app(neq, c_app(s, x)), 0) -> true [1] app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(c_app(filter, f), ys) [1] app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(null_app, ys) [1] app(c_app(neq, 0), 0) -> false [1] app(c_app(filter, c_app(neq, c_app(s, x'))), c_app(c_app(cons, 0), ys)) -> app(app(app(filtersub, true), c_app(neq, c_app(s, x'))), app(c_app(cons, 0), ys)) [2] app(c_app(filter, c_app(neq, c_app(s, x'))), c_app(c_app(cons, 0), ys)) -> app(app(app(filtersub, true), c_app(neq, c_app(s, x'))), app(null_app, ys)) [2] app(c_app(filter, c_app(c_app(filtersub, false), f')), c_app(c_app(cons, c_app(c_app(cons, y'), ys')), ys)) -> app(app(app(filtersub, app(app(filter, f'), ys')), c_app(c_app(filtersub, false), f')), app(c_app(cons, c_app(c_app(cons, y'), ys')), ys)) [2] app(c_app(filter, c_app(c_app(filtersub, false), f')), c_app(c_app(cons, c_app(c_app(cons, y'), ys')), ys)) -> app(app(app(filtersub, app(app(filter, f'), ys')), c_app(c_app(filtersub, false), f')), app(null_app, ys)) [2] app(c_app(filter, c_app(neq, 0)), c_app(c_app(cons, 0), ys)) -> app(app(app(filtersub, false), c_app(neq, 0)), app(c_app(cons, 0), ys)) [2] app(c_app(filter, c_app(neq, 0)), c_app(c_app(cons, 0), ys)) -> app(app(app(filtersub, false), c_app(neq, 0)), app(null_app, ys)) [2] app(c_app(filter, c_app(filter, f'')), c_app(c_app(cons, c_app(c_app(cons, y''), ys'')), ys)) -> app(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), c_app(filter, f'')), app(c_app(cons, c_app(c_app(cons, y''), ys'')), ys)) [2] app(c_app(filter, c_app(filter, f'')), c_app(c_app(cons, c_app(c_app(cons, y''), ys'')), ys)) -> app(app(app(filtersub, app(app(app(filtersub, app(f'', y'')), f''), app(app(cons, y''), ys''))), c_app(filter, f'')), app(null_app, ys)) [2] app(c_app(filter, c_app(neq, c_app(s, x''))), c_app(c_app(cons, c_app(s, y1)), ys)) -> app(app(app(filtersub, app(app(neq, x''), y1)), c_app(neq, c_app(s, x''))), app(c_app(cons, c_app(s, y1)), ys)) [2] app(c_app(filter, c_app(neq, c_app(s, x''))), c_app(c_app(cons, c_app(s, y1)), ys)) -> app(app(app(filtersub, app(app(neq, x''), y1)), c_app(neq, c_app(s, x''))), app(null_app, ys)) [2] app(c_app(filter, c_app(filter, f1)), c_app(c_app(cons, nil), ys)) -> app(app(app(filtersub, nil), c_app(filter, f1)), app(c_app(cons, nil), ys)) [2] app(c_app(filter, c_app(filter, f1)), c_app(c_app(cons, nil), ys)) -> app(app(app(filtersub, nil), c_app(filter, f1)), app(null_app, ys)) [2] app(c_app(filter, c_app(c_app(filtersub, true), f2)), c_app(c_app(cons, c_app(c_app(cons, y2), ys1)), ys)) -> app(app(app(filtersub, app(app(cons, y2), app(app(filter, f2), ys1))), c_app(c_app(filtersub, true), f2)), app(c_app(cons, c_app(c_app(cons, y2), ys1)), ys)) [2] app(c_app(filter, c_app(c_app(filtersub, true), f2)), c_app(c_app(cons, c_app(c_app(cons, y2), ys1)), ys)) -> app(app(app(filtersub, app(app(cons, y2), app(app(filter, f2), ys1))), c_app(c_app(filtersub, true), f2)), app(null_app, ys)) [2] app(c_app(filter, c_app(neq, 0)), c_app(c_app(cons, c_app(s, y3)), ys)) -> app(app(app(filtersub, true), c_app(neq, 0)), app(c_app(cons, c_app(s, y3)), ys)) [2] app(c_app(filter, c_app(neq, 0)), c_app(c_app(cons, c_app(s, y3)), ys)) -> app(app(app(filtersub, true), c_app(neq, 0)), app(null_app, ys)) [2] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, c_app(f, y)), f), app(c_app(cons, y), ys)) [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, c_app(f, y)), f), app(null_app, ys)) [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, null_app), f), app(c_app(cons, y), ys)) [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, null_app), f), app(null_app, ys)) [1] app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(c_app(neq, x), y) [1] app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(null_app, y) [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(c_app(cons, y), app(c_app(filter, f), ys)) [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(c_app(cons, y), app(null_app, ys)) [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(null_app, app(c_app(filter, f), ys)) [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(null_app, app(null_app, ys)) [1] app(c_app(neq, 0), c_app(s, y)) -> true [1] encArg(neq) -> neq [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s) -> s [0] encArg(true) -> true [0] encArg(filter) -> filter [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filtersub) -> filtersub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encArg(cons_nonzero) -> nonzero [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_neq -> neq [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s -> s [0] encode_true -> true [0] encode_filter -> filter [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filtersub -> filtersub [0] encode_nonzero -> nonzero [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_neq -> null_encode_neq [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s -> null_encode_s [0] encode_true -> null_encode_true [0] encode_filter -> null_encode_filter [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filtersub -> null_encode_filtersub [0] encode_nonzero -> null_encode_nonzero [0] app(v0, v1) -> null_app [0] The TRS has the following type information: nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app 0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app c_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero: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: filter => 4 neq => 6 0 => 0 s => 8 true => 9 filtersub => 5 false => 3 cons => 1 nil => 7 cons_nonzero => 2 null_encArg => 0 null_encode_app => 0 null_encode_neq => 0 null_encode_0 => 0 null_encode_false => 0 null_encode_s => 0 null_encode_true => 0 null_encode_filter => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filtersub => 0 null_encode_nonzero => 0 null_app => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 2 }-> app(app(app(5, app(app(app(5, app(f'', y'')), f''), app(app(1, y''), ys''))), 1 + 4 + f''), app(0, ys)) :|: z = 1 + 4 + (1 + 4 + f''), z' = 1 + (1 + 1 + (1 + (1 + 1 + y'') + ys'')) + ys, f'' >= 0, ys >= 0, y'' >= 0, ys'' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(app(5, app(f'', y'')), f''), app(app(1, y''), ys''))), 1 + 4 + f''), app(1 + 1 + (1 + (1 + 1 + y'') + ys''), ys)) :|: z = 1 + 4 + (1 + 4 + f''), z' = 1 + (1 + 1 + (1 + (1 + 1 + y'') + ys'')) + ys, f'' >= 0, ys >= 0, y'' >= 0, ys'' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(6, x''), y1)), 1 + 6 + (1 + 8 + x'')), app(0, ys)) :|: y1 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y1)) + ys, ys >= 0, z = 1 + 4 + (1 + 6 + (1 + 8 + x'')), x'' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(6, x''), y1)), 1 + 6 + (1 + 8 + x'')), app(1 + 1 + (1 + 8 + y1), ys)) :|: y1 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y1)) + ys, ys >= 0, z = 1 + 4 + (1 + 6 + (1 + 8 + x'')), x'' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(4, f'), ys')), 1 + (1 + 5 + 3) + f'), app(0, ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y') + ys')) + ys, z = 1 + 4 + (1 + (1 + 5 + 3) + f'), f' >= 0, ys >= 0, ys' >= 0, y' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(4, f'), ys')), 1 + (1 + 5 + 3) + f'), app(1 + 1 + (1 + (1 + 1 + y') + ys'), ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y') + ys')) + ys, z = 1 + 4 + (1 + (1 + 5 + 3) + f'), f' >= 0, ys >= 0, ys' >= 0, y' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(1, y2), app(app(4, f2), ys1))), 1 + (1 + 5 + 9) + f2), app(0, ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y2) + ys1)) + ys, z = 1 + 4 + (1 + (1 + 5 + 9) + f2), ys1 >= 0, ys >= 0, f2 >= 0, y2 >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(1, y2), app(app(4, f2), ys1))), 1 + (1 + 5 + 9) + f2), app(1 + 1 + (1 + (1 + 1 + y2) + ys1), ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y2) + ys1)) + ys, z = 1 + 4 + (1 + (1 + 5 + 9) + f2), ys1 >= 0, ys >= 0, f2 >= 0, y2 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + 0), app(0, ys)) :|: z = 1 + 4 + (1 + 6 + 0), ys >= 0, y3 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y3)) + ys app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + 0), app(1 + 1 + (1 + 8 + y3), ys)) :|: z = 1 + 4 + (1 + 6 + 0), ys >= 0, y3 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y3)) + ys app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + (1 + 8 + x')), app(0, ys)) :|: z' = 1 + (1 + 1 + 0) + ys, z = 1 + 4 + (1 + 6 + (1 + 8 + x')), ys >= 0, x' >= 0 app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + (1 + 8 + x')), app(1 + 1 + 0, ys)) :|: z' = 1 + (1 + 1 + 0) + ys, z = 1 + 4 + (1 + 6 + (1 + 8 + x')), ys >= 0, x' >= 0 app(z, z') -{ 2 }-> app(app(app(5, 7), 1 + 4 + f1), app(0, ys)) :|: ys >= 0, z' = 1 + (1 + 1 + 7) + ys, f1 >= 0, z = 1 + 4 + (1 + 4 + f1) app(z, z') -{ 2 }-> app(app(app(5, 7), 1 + 4 + f1), app(1 + 1 + 7, ys)) :|: ys >= 0, z' = 1 + (1 + 1 + 7) + ys, f1 >= 0, z = 1 + 4 + (1 + 4 + f1) app(z, z') -{ 2 }-> app(app(app(5, 3), 1 + 6 + 0), app(0, ys)) :|: z' = 1 + (1 + 1 + 0) + ys, z = 1 + 4 + (1 + 6 + 0), ys >= 0 app(z, z') -{ 2 }-> app(app(app(5, 3), 1 + 6 + 0), app(1 + 1 + 0, ys)) :|: z' = 1 + (1 + 1 + 0) + ys, z = 1 + 4 + (1 + 6 + 0), ys >= 0 app(z, z') -{ 1 }-> app(app(app(5, 0), f), app(0, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + 4 + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(app(5, 0), f), app(1 + 1 + y, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + 4 + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(app(5, 1 + f + y), f), app(0, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + 4 + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(app(5, 1 + f + y), f), app(1 + 1 + y, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + 4 + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, y) :|: x >= 0, y >= 0, z = 1 + 6 + (1 + 8 + x), z' = 1 + 8 + y app(z, z') -{ 1 }-> app(0, ys) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 3) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, app(0, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 9) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, app(1 + 4 + f, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 9) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(1 + 6 + x, y) :|: x >= 0, y >= 0, z = 1 + 6 + (1 + 8 + x), z' = 1 + 8 + y app(z, z') -{ 1 }-> app(1 + 4 + f, ys) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 3) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(1 + 1 + y, app(0, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 9) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(1 + 1 + y, app(1 + 4 + f, ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 9) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> 9 :|: x >= 0, z = 1 + 6 + (1 + 8 + x), z' = 0 app(z, z') -{ 1 }-> 9 :|: y >= 0, z = 1 + 6 + 0, z' = 1 + 8 + y app(z, z') -{ 1 }-> 7 :|: f >= 0, z = 1 + 4 + f, z' = 7 app(z, z') -{ 1 }-> 3 :|: z = 1 + 6 + 0, z' = 0 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> nonzero :|: z = 2 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 9 :|: z = 9 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 3 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 4 :|: encode_filter -{ 0 }-> 0 :|: encode_filtersub -{ 0 }-> 5 :|: encode_filtersub -{ 0 }-> 0 :|: encode_neq -{ 0 }-> 6 :|: encode_neq -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_nonzero -{ 0 }-> nonzero :|: encode_nonzero -{ 0 }-> 0 :|: encode_s -{ 0 }-> 8 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 9 :|: encode_true -{ 0 }-> 0 :|: nonzero -{ 1 }-> app(4, 0) :|: nonzero -{ 1 }-> app(4, 1 + 6 + 0) :|: ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 2 }-> app(app(app(5, app(app(app(5, app(z - 10, y'')), z - 10), app(app(1, y''), ys''))), 1 + 4 + (z - 10)), app(0, ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y'') + ys'')) + ys, z - 10 >= 0, ys >= 0, y'' >= 0, ys'' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(app(5, app(z - 10, y'')), z - 10), app(app(1, y''), ys''))), 1 + 4 + (z - 10)), app(1 + 1 + (1 + (1 + 1 + y'') + ys''), ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y'') + ys'')) + ys, z - 10 >= 0, ys >= 0, y'' >= 0, ys'' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(6, z - 21), y1)), 1 + 6 + (1 + 8 + (z - 21))), app(0, ys)) :|: y1 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y1)) + ys, ys >= 0, z - 21 >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(6, z - 21), y1)), 1 + 6 + (1 + 8 + (z - 21))), app(1 + 1 + (1 + 8 + y1), ys)) :|: y1 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y1)) + ys, ys >= 0, z - 21 >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(4, z - 15), ys')), 1 + (1 + 5 + 3) + (z - 15)), app(0, ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y') + ys')) + ys, z - 15 >= 0, ys >= 0, ys' >= 0, y' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(4, z - 15), ys')), 1 + (1 + 5 + 3) + (z - 15)), app(1 + 1 + (1 + (1 + 1 + y') + ys'), ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y') + ys')) + ys, z - 15 >= 0, ys >= 0, ys' >= 0, y' >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(1, y2), app(app(4, z - 21), ys1))), 1 + (1 + 5 + 9) + (z - 21)), app(0, ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y2) + ys1)) + ys, ys1 >= 0, ys >= 0, z - 21 >= 0, y2 >= 0 app(z, z') -{ 2 }-> app(app(app(5, app(app(1, y2), app(app(4, z - 21), ys1))), 1 + (1 + 5 + 9) + (z - 21)), app(1 + 1 + (1 + (1 + 1 + y2) + ys1), ys)) :|: z' = 1 + (1 + 1 + (1 + (1 + 1 + y2) + ys1)) + ys, ys1 >= 0, ys >= 0, z - 21 >= 0, y2 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + 0), app(0, ys)) :|: z = 1 + 4 + (1 + 6 + 0), ys >= 0, y3 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y3)) + ys app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + 0), app(1 + 1 + (1 + 8 + y3), ys)) :|: z = 1 + 4 + (1 + 6 + 0), ys >= 0, y3 >= 0, z' = 1 + (1 + 1 + (1 + 8 + y3)) + ys app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + (1 + 8 + (z - 21))), app(0, z' - 3)) :|: z' - 3 >= 0, z - 21 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 9), 1 + 6 + (1 + 8 + (z - 21))), app(1 + 1 + 0, z' - 3)) :|: z' - 3 >= 0, z - 21 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 7), 1 + 4 + (z - 10)), app(0, z' - 10)) :|: z' - 10 >= 0, z - 10 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 7), 1 + 4 + (z - 10)), app(1 + 1 + 7, z' - 10)) :|: z' - 10 >= 0, z - 10 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 3), 1 + 6 + 0), app(0, z' - 3)) :|: z = 1 + 4 + (1 + 6 + 0), z' - 3 >= 0 app(z, z') -{ 2 }-> app(app(app(5, 3), 1 + 6 + 0), app(1 + 1 + 0, z' - 3)) :|: z = 1 + 4 + (1 + 6 + 0), z' - 3 >= 0 app(z, z') -{ 1 }-> app(app(app(5, 0), z - 5), app(0, ys)) :|: z - 5 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(app(5, 0), z - 5), app(1 + 1 + y, ys)) :|: z - 5 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(app(5, 1 + (z - 5) + y), z - 5), app(0, ys)) :|: z - 5 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(app(5, 1 + (z - 5) + y), z - 5), app(1 + 1 + y, ys)) :|: z - 5 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, ys) :|: z - 10 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, app(0, ys)) :|: z - 16 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, app(1 + 4 + (z - 16), ys)) :|: z - 16 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(0, z' - 9) :|: z - 16 >= 0, z' - 9 >= 0 app(z, z') -{ 1 }-> app(1 + 6 + (z - 16), z' - 9) :|: z - 16 >= 0, z' - 9 >= 0 app(z, z') -{ 1 }-> app(1 + 4 + (z - 10), ys) :|: z - 10 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(1 + 1 + y, app(0, ys)) :|: z - 16 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(1 + 1 + y, app(1 + 4 + (z - 16), ys)) :|: z - 16 >= 0, ys >= 0, y >= 0, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> 9 :|: z - 16 >= 0, z' = 0 app(z, z') -{ 1 }-> 9 :|: z' - 9 >= 0, z = 1 + 6 + 0 app(z, z') -{ 1 }-> 7 :|: z - 5 >= 0, z' = 7 app(z, z') -{ 1 }-> 3 :|: z = 1 + 6 + 0, z' = 0 app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 app(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> nonzero :|: z = 2 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 9 :|: z = 9 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 3 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 4 :|: encode_filter -{ 0 }-> 0 :|: encode_filtersub -{ 0 }-> 5 :|: encode_filtersub -{ 0 }-> 0 :|: encode_neq -{ 0 }-> 6 :|: encode_neq -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_nonzero -{ 0 }-> nonzero :|: encode_nonzero -{ 0 }-> 0 :|: encode_s -{ 0 }-> 8 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 9 :|: encode_true -{ 0 }-> 0 :|: nonzero -{ 1 }-> app(4, 0) :|: nonzero -{ 1 }-> app(4, 1 + 6 + 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_neq -> null_encode_neq [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s -> null_encode_s [0] encode_true -> null_encode_true [0] encode_filter -> null_encode_filter [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filtersub -> null_encode_filtersub [0] encode_nonzero -> null_encode_nonzero [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_neq, null_encode_0, null_encode_false, null_encode_s, null_encode_true, null_encode_filter, null_encode_nil, null_encode_cons, null_encode_filtersub, null_encode_nonzero, 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: nonzero -> app(filter, app(neq, 0)) [1] app(c_app(neq, c_app(s, x)), 0) -> true [1] app(c_app(c_app(filtersub, false), f), c_app(c_app(cons, y), ys)) -> app(app(filter, f), ys) [1] app(c_app(neq, 0), 0) -> false [1] app(c_app(filter, f), c_app(c_app(cons, y), ys)) -> app(app(app(filtersub, app(f, y)), f), app(app(cons, y), ys)) [1] app(c_app(neq, c_app(s, x)), c_app(s, y)) -> app(app(neq, x), y) [1] app(c_app(filter, f), nil) -> nil [1] app(c_app(c_app(filtersub, true), f), c_app(c_app(cons, y), ys)) -> app(app(cons, y), app(app(filter, f), ys)) [1] app(c_app(neq, 0), c_app(s, y)) -> true [1] encArg(neq) -> neq [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s) -> s [0] encArg(true) -> true [0] encArg(filter) -> filter [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(filtersub) -> filtersub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encArg(cons_nonzero) -> nonzero [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_neq -> neq [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s -> s [0] encode_true -> true [0] encode_filter -> filter [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_filtersub -> filtersub [0] encode_nonzero -> nonzero [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_neq -> null_encode_neq [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s -> null_encode_s [0] encode_true -> null_encode_true [0] encode_filter -> null_encode_filter [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_filtersub -> null_encode_filtersub [0] encode_nonzero -> null_encode_nonzero [0] app(v0, v1) -> null_app [0] The TRS has the following type information: nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app 0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app c_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app cons_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app -> filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encArg :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_neq :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_0 :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_false :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_s :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_true :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_filter :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_nil :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_cons :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_filtersub :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_encode_nonzero :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero:null_app null_app :: filter:neq:0:s:c_app:true:filtersub:false:cons:nil:cons_app:cons_nonzero:null_encArg:null_encode_app:null_encode_neq:null_encode_0:null_encode_false:null_encode_s:null_encode_true:null_encode_filter:null_encode_nil:null_encode_cons:null_encode_filtersub:null_encode_nonzero: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: filter => 4 neq => 6 0 => 0 s => 8 true => 9 filtersub => 5 false => 3 cons => 1 nil => 7 cons_nonzero => 2 null_encArg => 0 null_encode_app => 0 null_encode_neq => 0 null_encode_0 => 0 null_encode_false => 0 null_encode_s => 0 null_encode_true => 0 null_encode_filter => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_filtersub => 0 null_encode_nonzero => 0 null_app => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> app(app(app(5, app(f, y)), f), app(app(1, y), ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + 4 + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(6, x), y) :|: x >= 0, y >= 0, z = 1 + 6 + (1 + 8 + x), z' = 1 + 8 + y app(z, z') -{ 1 }-> app(app(4, f), ys) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 3) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> app(app(1, y), app(app(4, f), ys)) :|: f >= 0, ys >= 0, y >= 0, z = 1 + (1 + 5 + 9) + f, z' = 1 + (1 + 1 + y) + ys app(z, z') -{ 1 }-> 9 :|: x >= 0, z = 1 + 6 + (1 + 8 + x), z' = 0 app(z, z') -{ 1 }-> 9 :|: y >= 0, z = 1 + 6 + 0, z' = 1 + 8 + y app(z, z') -{ 1 }-> 7 :|: f >= 0, z = 1 + 4 + f, z' = 7 app(z, z') -{ 1 }-> 3 :|: z = 1 + 6 + 0, z' = 0 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> nonzero :|: z = 2 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 9 :|: z = 9 encArg(z) -{ 0 }-> 8 :|: z = 8 encArg(z) -{ 0 }-> 7 :|: z = 7 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_0 -{ 0 }-> 0 :|: encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_false -{ 0 }-> 3 :|: encode_false -{ 0 }-> 0 :|: encode_filter -{ 0 }-> 4 :|: encode_filter -{ 0 }-> 0 :|: encode_filtersub -{ 0 }-> 5 :|: encode_filtersub -{ 0 }-> 0 :|: encode_neq -{ 0 }-> 6 :|: encode_neq -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 7 :|: encode_nil -{ 0 }-> 0 :|: encode_nonzero -{ 0 }-> nonzero :|: encode_nonzero -{ 0 }-> 0 :|: encode_s -{ 0 }-> 8 :|: encode_s -{ 0 }-> 0 :|: encode_true -{ 0 }-> 9 :|: encode_true -{ 0 }-> 0 :|: nonzero -{ 1 }-> app(4, app(6, 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(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(neq) -> c ENCARG(0) -> c1 ENCARG(false) -> c2 ENCARG(s) -> c3 ENCARG(true) -> c4 ENCARG(filter) -> c5 ENCARG(nil) -> c6 ENCARG(cons) -> c7 ENCARG(filtersub) -> c8 ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_NEQ -> c12 ENCODE_0 -> c13 ENCODE_FALSE -> c14 ENCODE_S -> c15 ENCODE_TRUE -> c16 ENCODE_FILTER -> c17 ENCODE_NIL -> c18 ENCODE_CONS -> c19 ENCODE_FILTERSUB -> c20 ENCODE_NONZERO -> c21(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1), APP(neq, z0)) APP(app(filter, z0), nil) -> c26 APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(app(filtersub, app(z0, z1)), z0), APP(filtersub, app(z0, z1)), APP(z0, z1), APP(app(cons, z1), z2), APP(cons, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2), APP(filter, z0)) NONZERO -> c30(APP(filter, app(neq, 0)), APP(neq, 0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1), APP(neq, z0)) APP(app(filter, z0), nil) -> c26 APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(app(filtersub, app(z0, z1)), z0), APP(filtersub, app(z0, z1)), APP(z0, z1), APP(app(cons, z1), z2), APP(cons, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2), APP(filter, z0)) NONZERO -> c30(APP(filter, app(neq, 0)), APP(neq, 0)) K tuples:none Defined Rule Symbols: app_2, nonzero, encArg_1, encode_app_2, encode_neq, encode_0, encode_false, encode_s, encode_true, encode_filter, encode_nil, encode_cons, encode_filtersub, encode_nonzero Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_NEQ, ENCODE_0, ENCODE_FALSE, ENCODE_S, ENCODE_TRUE, ENCODE_FILTER, ENCODE_NIL, ENCODE_CONS, ENCODE_FILTERSUB, ENCODE_NONZERO, APP_2, NONZERO Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8, c9_3, c10_1, c11_3, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21_1, c22, c23, c24, c25_2, c26, c27_6, c28_4, c29_2, c30_2 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_NONZERO -> c21(NONZERO) Removed 18 trailing nodes: ENCARG(cons) -> c7 ENCODE_TRUE -> c16 ENCODE_FILTER -> c17 ENCODE_0 -> c13 ENCARG(nil) -> c6 ENCARG(s) -> c3 ENCODE_FALSE -> c14 ENCARG(neq) -> c ENCODE_NEQ -> c12 ENCARG(filter) -> c5 ENCARG(false) -> c2 ENCODE_NIL -> c18 ENCODE_CONS -> c19 ENCARG(filtersub) -> c8 ENCARG(0) -> c1 ENCARG(true) -> c4 ENCODE_FILTERSUB -> c20 ENCODE_S -> c15 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1), APP(neq, z0)) APP(app(filter, z0), nil) -> c26 APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(app(filtersub, app(z0, z1)), z0), APP(filtersub, app(z0, z1)), APP(z0, z1), APP(app(cons, z1), z2), APP(cons, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2), APP(filter, z0)) NONZERO -> c30(APP(filter, app(neq, 0)), APP(neq, 0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1), APP(neq, z0)) APP(app(filter, z0), nil) -> c26 APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(app(filtersub, app(z0, z1)), z0), APP(filtersub, app(z0, z1)), APP(z0, z1), APP(app(cons, z1), z2), APP(cons, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(cons, z1), app(app(filter, z0), z2)), APP(cons, z1), APP(app(filter, z0), z2), APP(filter, z0)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2), APP(filter, z0)) NONZERO -> c30(APP(filter, app(neq, 0)), APP(neq, 0)) K tuples:none Defined Rule Symbols: app_2, nonzero, encArg_1, encode_app_2, encode_neq, encode_0, encode_false, encode_s, encode_true, encode_filter, encode_nil, encode_cons, encode_filtersub, encode_nonzero Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2, NONZERO Compound Symbols: c9_3, c10_1, c11_3, c22, c23, c24, c25_2, c26, c27_6, c28_4, c29_2, c30_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) ENCODE_APP(z0, z1) -> c11(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 K tuples:none Defined Rule Symbols: app_2, nonzero, encArg_1, encode_app_2, encode_neq, encode_0, encode_false, encode_s, encode_true, encode_filter, encode_nil, encode_cons, encode_filtersub, encode_nonzero Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2, NONZERO Compound Symbols: c9_3, c10_1, c11_3, c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c30 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 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(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 K tuples:none Defined Rule Symbols: app_2, nonzero, encArg_1, encode_app_2, encode_neq, encode_0, encode_false, encode_s, encode_true, encode_filter, encode_nil, encode_cons, encode_filtersub, encode_nonzero Defined Pair Symbols: ENCARG_1, APP_2, NONZERO, ENCODE_APP_2 Compound Symbols: c9_3, c10_1, c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c30, 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(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 K tuples:none Defined Rule Symbols: app_2, nonzero, encArg_1, encode_app_2, encode_neq, encode_0, encode_false, encode_s, encode_true, encode_filter, encode_nil, encode_cons, encode_filtersub, encode_nonzero Defined Pair Symbols: ENCARG_1, APP_2, NONZERO, ENCODE_APP_2 Compound Symbols: c9_3, c10_1, c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c30, 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_neq -> neq encode_0 -> 0 encode_false -> false encode_s -> s encode_true -> true encode_filter -> filter encode_nil -> nil encode_cons -> cons encode_filtersub -> filtersub encode_nonzero -> nonzero ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: ENCARG_1, APP_2, NONZERO, ENCODE_APP_2 Compound Symbols: c9_3, c10_1, c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c30, c_1 ---------------------------------------- (47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. NONZERO -> c30 We considered the (Usable) Rules:none And the Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(APP(x_1, x_2)) = 0 POL(ENCARG(x_1)) = [1] + x_1 POL(ENCODE_APP(x_1, x_2)) = 0 POL(NONZERO) = [1] POL(app(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c22) = 0 POL(c23) = 0 POL(c24) = 0 POL(c25(x_1)) = x_1 POL(c26) = 0 POL(c27(x_1, x_2)) = x_1 + x_2 POL(c28(x_1)) = x_1 POL(c29(x_1)) = x_1 POL(c30) = 0 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(cons) = [1] POL(cons_app(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_nonzero) = [1] POL(encArg(x_1)) = [1] + x_1 POL(false) = [1] POL(filter) = [1] POL(filtersub) = [1] POL(neq) = [1] POL(nil) = [1] POL(nonzero) = [1] POL(s) = [1] POL(true) = 0 ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_nonzero) -> c10(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) K tuples: NONZERO -> c30 Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: ENCARG_1, APP_2, NONZERO, ENCODE_APP_2 Compound Symbols: c9_3, c10_1, c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c30, c_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c9(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, neq)) -> c9(APP(encArg(x0), neq), ENCARG(x0), ENCARG(neq)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, s)) -> c9(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filtersub)) -> c9(APP(encArg(x0), filtersub), ENCARG(x0), ENCARG(filtersub)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0), ENCARG(cons_nonzero)) ENCARG(cons_app(neq, x1)) -> c9(APP(neq, encArg(x1)), ENCARG(neq), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(APP(filtersub, encArg(x1)), ENCARG(filtersub), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(cons_nonzero), ENCARG(x1)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: ENCARG(cons_nonzero) -> c10(NONZERO) APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) NONZERO -> c30 ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, neq)) -> c9(APP(encArg(x0), neq), ENCARG(x0), ENCARG(neq)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, s)) -> c9(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filtersub)) -> c9(APP(encArg(x0), filtersub), ENCARG(x0), ENCARG(filtersub)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0), ENCARG(cons_nonzero)) ENCARG(cons_app(neq, x1)) -> c9(APP(neq, encArg(x1)), ENCARG(neq), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(APP(filtersub, encArg(x1)), ENCARG(filtersub), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(cons_nonzero), ENCARG(x1)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) K tuples: NONZERO -> c30 Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: ENCARG_1, APP_2, NONZERO, ENCODE_APP_2 Compound Symbols: c10_1, c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c30, c_1, c9_3 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCARG(cons_nonzero) -> c10(NONZERO) NONZERO -> c30 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, neq)) -> c9(APP(encArg(x0), neq), ENCARG(x0), ENCARG(neq)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0), ENCARG(0)) ENCARG(cons_app(x0, false)) -> c9(APP(encArg(x0), false), ENCARG(x0), ENCARG(false)) ENCARG(cons_app(x0, s)) -> c9(APP(encArg(x0), s), ENCARG(x0), ENCARG(s)) ENCARG(cons_app(x0, true)) -> c9(APP(encArg(x0), true), ENCARG(x0), ENCARG(true)) ENCARG(cons_app(x0, filter)) -> c9(APP(encArg(x0), filter), ENCARG(x0), ENCARG(filter)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c9(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, filtersub)) -> c9(APP(encArg(x0), filtersub), ENCARG(x0), ENCARG(filtersub)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0), ENCARG(cons_nonzero)) ENCARG(cons_app(neq, x1)) -> c9(APP(neq, encArg(x1)), ENCARG(neq), ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(APP(0, encArg(x1)), ENCARG(0), ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(APP(false, encArg(x1)), ENCARG(false), ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(APP(s, encArg(x1)), ENCARG(s), ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(APP(true, encArg(x1)), ENCARG(true), ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(APP(filter, encArg(x1)), ENCARG(filter), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(APP(filtersub, encArg(x1)), ENCARG(filtersub), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(cons_nonzero), ENCARG(x1)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c_1, c9_3 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 36 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c23, c24, c26, c25_1, c27_2, c28_1, c29_1, c_1, c9_3, c9_1, c9_2 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, z0), app(app(cons, z1), z2)) -> c27(APP(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)), APP(z0, z1)) by APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(app(filtersub, nil), app(filter, z0)), app(app(cons, nil), x2)), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(cons, z1), app(app(filter, z0), z2))), app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(app(filtersub, nil), app(filter, z0)), app(app(cons, nil), x2)), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(cons, z1), app(app(filter, z0), z2))), app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(app(filtersub, nil), app(filter, z0)), app(app(cons, nil), x2)), APP(app(filter, z0), nil)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(cons, z1), app(app(filter, z0), z2))), app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c_1, c9_3, c9_1, c9_2, c27_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c_1, c9_3, c9_1, c9_2, c27_2, c27_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, neq) -> c(APP(encArg(x0), neq)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filtersub) -> c(APP(encArg(x0), filtersub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(neq, x1) -> c(APP(neq, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filtersub, x1) -> c(APP(filtersub, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, neq) -> c(APP(encArg(x0), neq)) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, filtersub) -> c(APP(encArg(x0), filtersub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(neq, x1) -> c(APP(neq, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(filtersub, x1) -> c(APP(filtersub, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_3, c9_1, c9_2, c27_2, c27_1, c_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing nodes: ENCODE_APP(x0, true) -> c(APP(encArg(x0), true)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, s) -> c(APP(encArg(x0), s)) ENCODE_APP(false, x1) -> c(APP(false, encArg(x1))) ENCODE_APP(true, x1) -> c(APP(true, encArg(x1))) ENCODE_APP(filter, x1) -> c(APP(filter, encArg(x1))) ENCODE_APP(0, x1) -> c(APP(0, encArg(x1))) ENCODE_APP(x0, neq) -> c(APP(encArg(x0), neq)) ENCODE_APP(neq, x1) -> c(APP(neq, encArg(x1))) ENCODE_APP(x0, filtersub) -> c(APP(encArg(x0), filtersub)) ENCODE_APP(filtersub, x1) -> c(APP(filtersub, encArg(x1))) ENCODE_APP(x0, false) -> c(APP(encArg(x0), false)) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(s, x1) -> c(APP(s, encArg(x1))) ENCODE_APP(x0, filter) -> c(APP(encArg(x0), filter)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_3, c9_1, c9_2, c27_2, c27_1, c_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c9(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(APP(encArg(x0), app(neq, encArg(x2))), ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(APP(encArg(x0), app(filtersub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(APP(neq, app(encArg(x1), encArg(x2))), ENCARG(neq), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(APP(filtersub, app(encArg(x1), encArg(x2))), ENCARG(filtersub), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_nonzero), ENCARG(cons_app(x1, x2))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(APP(encArg(x0), app(neq, encArg(x2))), ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(APP(encArg(x0), app(0, encArg(x2))), ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(APP(encArg(x0), app(false, encArg(x2))), ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(APP(encArg(x0), app(true, encArg(x2))), ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(APP(encArg(x0), app(filter, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(APP(encArg(x0), app(filtersub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(APP(neq, app(encArg(x1), encArg(x2))), ENCARG(neq), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(APP(0, app(encArg(x1), encArg(x2))), ENCARG(0), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(APP(false, app(encArg(x1), encArg(x2))), ENCARG(false), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(APP(s, app(encArg(x1), encArg(x2))), ENCARG(s), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(APP(true, app(encArg(x1), encArg(x2))), ENCARG(true), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(APP(filter, app(encArg(x1), encArg(x2))), ENCARG(filter), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(APP(filtersub, app(encArg(x1), encArg(x2))), ENCARG(filtersub), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_nonzero), ENCARG(cons_app(x1, x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_3, c9_1, c9_2, c27_2, c27_1, c_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 27 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_3, c9_1, c9_2, c27_2, c27_1, c_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c9(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(APP(app(encArg(x0), encArg(x1)), neq), ENCARG(cons_app(x0, x1)), ENCARG(neq)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(APP(app(encArg(x0), encArg(x1)), filtersub), ENCARG(cons_app(x0, x1)), ENCARG(filtersub)) ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c9(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1))), ENCARG(cons_app(x0, x1)), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1)), ENCARG(cons_nonzero)) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(APP(app(filtersub, encArg(x1)), encArg(x2)), ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(APP(app(encArg(x0), encArg(x1)), neq), ENCARG(cons_app(x0, x1)), ENCARG(neq)) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1)), ENCARG(0)) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(APP(app(encArg(x0), encArg(x1)), false), ENCARG(cons_app(x0, x1)), ENCARG(false)) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(APP(app(encArg(x0), encArg(x1)), s), ENCARG(cons_app(x0, x1)), ENCARG(s)) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(APP(app(encArg(x0), encArg(x1)), true), ENCARG(cons_app(x0, x1)), ENCARG(true)) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(APP(app(encArg(x0), encArg(x1)), filter), ENCARG(cons_app(x0, x1)), ENCARG(filter)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(APP(app(encArg(x0), encArg(x1)), filtersub), ENCARG(cons_app(x0, x1)), ENCARG(filtersub)) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1)), ENCARG(cons_nonzero)) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(APP(app(0, encArg(x1)), encArg(x2)), ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(APP(app(false, encArg(x1)), encArg(x2)), ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(APP(app(s, encArg(x1)), encArg(x2)), ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(APP(app(true, encArg(x1)), encArg(x2)), ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(APP(app(filtersub, encArg(x1)), encArg(x2)), ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 24 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, 0)) -> c9(APP(encArg(x0), 0), ENCARG(x0)) by ENCARG(cons_app(neq, 0)) -> c9(APP(neq, 0), ENCARG(neq)) ENCARG(cons_app(0, 0)) -> c9(APP(0, 0), ENCARG(0)) ENCARG(cons_app(false, 0)) -> c9(APP(false, 0), ENCARG(false)) ENCARG(cons_app(s, 0)) -> c9(APP(s, 0), ENCARG(s)) ENCARG(cons_app(true, 0)) -> c9(APP(true, 0), ENCARG(true)) ENCARG(cons_app(filter, 0)) -> c9(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(nil, 0)) -> c9(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c9(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filtersub, 0)) -> c9(APP(filtersub, 0), ENCARG(filtersub)) ENCARG(cons_app(cons_app(z0, z1), 0)) -> c9(APP(app(encArg(z0), encArg(z1)), 0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0), ENCARG(cons_nonzero)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(neq, 0)) -> c9(APP(neq, 0), ENCARG(neq)) ENCARG(cons_app(0, 0)) -> c9(APP(0, 0), ENCARG(0)) ENCARG(cons_app(false, 0)) -> c9(APP(false, 0), ENCARG(false)) ENCARG(cons_app(s, 0)) -> c9(APP(s, 0), ENCARG(s)) ENCARG(cons_app(true, 0)) -> c9(APP(true, 0), ENCARG(true)) ENCARG(cons_app(filter, 0)) -> c9(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(nil, 0)) -> c9(APP(nil, 0), ENCARG(nil)) ENCARG(cons_app(cons, 0)) -> c9(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(filtersub, 0)) -> c9(APP(filtersub, 0), ENCARG(filtersub)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0), ENCARG(cons_nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(filtersub, 0)) -> c9(APP(filtersub, 0), ENCARG(filtersub)) ENCARG(cons_app(neq, 0)) -> c9(APP(neq, 0), ENCARG(neq)) ENCARG(cons_app(filter, 0)) -> c9(APP(filter, 0), ENCARG(filter)) ENCARG(cons_app(cons, 0)) -> c9(APP(cons, 0), ENCARG(cons)) ENCARG(cons_app(s, 0)) -> c9(APP(s, 0), ENCARG(s)) ENCARG(cons_app(true, 0)) -> c9(APP(true, 0), ENCARG(true)) ENCARG(cons_app(0, 0)) -> c9(APP(0, 0), ENCARG(0)) ENCARG(cons_app(false, 0)) -> c9(APP(false, 0), ENCARG(false)) ENCARG(cons_app(nil, 0)) -> c9(APP(nil, 0), ENCARG(nil)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0), ENCARG(cons_nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c9(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(neq, nil)) -> c9(APP(neq, nil), ENCARG(neq)) ENCARG(cons_app(0, nil)) -> c9(APP(0, nil), ENCARG(0)) ENCARG(cons_app(false, nil)) -> c9(APP(false, nil), ENCARG(false)) ENCARG(cons_app(s, nil)) -> c9(APP(s, nil), ENCARG(s)) ENCARG(cons_app(true, nil)) -> c9(APP(true, nil), ENCARG(true)) ENCARG(cons_app(filter, nil)) -> c9(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(nil, nil)) -> c9(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c9(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filtersub, nil)) -> c9(APP(filtersub, nil), ENCARG(filtersub)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c9(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil), ENCARG(cons_nonzero)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(neq, nil)) -> c9(APP(neq, nil), ENCARG(neq)) ENCARG(cons_app(0, nil)) -> c9(APP(0, nil), ENCARG(0)) ENCARG(cons_app(false, nil)) -> c9(APP(false, nil), ENCARG(false)) ENCARG(cons_app(s, nil)) -> c9(APP(s, nil), ENCARG(s)) ENCARG(cons_app(true, nil)) -> c9(APP(true, nil), ENCARG(true)) ENCARG(cons_app(filter, nil)) -> c9(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(nil, nil)) -> c9(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c9(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(filtersub, nil)) -> c9(APP(filtersub, nil), ENCARG(filtersub)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil), ENCARG(cons_nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(filter, nil)) -> c9(APP(filter, nil), ENCARG(filter)) ENCARG(cons_app(false, nil)) -> c9(APP(false, nil), ENCARG(false)) ENCARG(cons_app(true, nil)) -> c9(APP(true, nil), ENCARG(true)) ENCARG(cons_app(0, nil)) -> c9(APP(0, nil), ENCARG(0)) ENCARG(cons_app(neq, nil)) -> c9(APP(neq, nil), ENCARG(neq)) ENCARG(cons_app(nil, nil)) -> c9(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c9(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(s, nil)) -> c9(APP(s, nil), ENCARG(s)) ENCARG(cons_app(filtersub, nil)) -> c9(APP(filtersub, nil), ENCARG(filtersub)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil), ENCARG(cons_nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), nonzero), ENCARG(x0)) by ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), app(filter, app(neq, 0))), ENCARG(x0)) ENCARG(cons_app(neq, cons_nonzero)) -> c9(APP(neq, nonzero), ENCARG(neq)) ENCARG(cons_app(0, cons_nonzero)) -> c9(APP(0, nonzero), ENCARG(0)) ENCARG(cons_app(false, cons_nonzero)) -> c9(APP(false, nonzero), ENCARG(false)) ENCARG(cons_app(s, cons_nonzero)) -> c9(APP(s, nonzero), ENCARG(s)) ENCARG(cons_app(true, cons_nonzero)) -> c9(APP(true, nonzero), ENCARG(true)) ENCARG(cons_app(filter, cons_nonzero)) -> c9(APP(filter, nonzero), ENCARG(filter)) ENCARG(cons_app(nil, cons_nonzero)) -> c9(APP(nil, nonzero), ENCARG(nil)) ENCARG(cons_app(cons, cons_nonzero)) -> c9(APP(cons, nonzero), ENCARG(cons)) ENCARG(cons_app(filtersub, cons_nonzero)) -> c9(APP(filtersub, nonzero), ENCARG(filtersub)) ENCARG(cons_app(cons_app(z0, z1), cons_nonzero)) -> c9(APP(app(encArg(z0), encArg(z1)), nonzero), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero), ENCARG(cons_nonzero)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), app(filter, app(neq, 0))), ENCARG(x0)) ENCARG(cons_app(neq, cons_nonzero)) -> c9(APP(neq, nonzero), ENCARG(neq)) ENCARG(cons_app(0, cons_nonzero)) -> c9(APP(0, nonzero), ENCARG(0)) ENCARG(cons_app(false, cons_nonzero)) -> c9(APP(false, nonzero), ENCARG(false)) ENCARG(cons_app(s, cons_nonzero)) -> c9(APP(s, nonzero), ENCARG(s)) ENCARG(cons_app(true, cons_nonzero)) -> c9(APP(true, nonzero), ENCARG(true)) ENCARG(cons_app(filter, cons_nonzero)) -> c9(APP(filter, nonzero), ENCARG(filter)) ENCARG(cons_app(nil, cons_nonzero)) -> c9(APP(nil, nonzero), ENCARG(nil)) ENCARG(cons_app(cons, cons_nonzero)) -> c9(APP(cons, nonzero), ENCARG(cons)) ENCARG(cons_app(filtersub, cons_nonzero)) -> c9(APP(filtersub, nonzero), ENCARG(filtersub)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero), ENCARG(cons_nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCARG(cons_app(0, cons_nonzero)) -> c9(APP(0, nonzero), ENCARG(0)) ENCARG(cons_app(filtersub, cons_nonzero)) -> c9(APP(filtersub, nonzero), ENCARG(filtersub)) ENCARG(cons_app(neq, cons_nonzero)) -> c9(APP(neq, nonzero), ENCARG(neq)) ENCARG(cons_app(nil, cons_nonzero)) -> c9(APP(nil, nonzero), ENCARG(nil)) ENCARG(cons_app(false, cons_nonzero)) -> c9(APP(false, nonzero), ENCARG(false)) ENCARG(cons_app(filter, cons_nonzero)) -> c9(APP(filter, nonzero), ENCARG(filter)) ENCARG(cons_app(cons, cons_nonzero)) -> c9(APP(cons, nonzero), ENCARG(cons)) ENCARG(cons_app(true, cons_nonzero)) -> c9(APP(true, nonzero), ENCARG(true)) ENCARG(cons_app(s, cons_nonzero)) -> c9(APP(s, nonzero), ENCARG(s)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(APP(encArg(x0), app(filter, app(neq, 0))), ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero), ENCARG(cons_nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c9_2, c27_2, c27_1, c_1, c9_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_nonzero, x1)) -> c9(APP(nonzero, encArg(x1)), ENCARG(x1)) by ENCARG(cons_app(cons_nonzero, neq)) -> c9(APP(nonzero, neq), ENCARG(neq)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0), ENCARG(0)) ENCARG(cons_app(cons_nonzero, false)) -> c9(APP(nonzero, false), ENCARG(false)) ENCARG(cons_app(cons_nonzero, s)) -> c9(APP(nonzero, s), ENCARG(s)) ENCARG(cons_app(cons_nonzero, true)) -> c9(APP(nonzero, true), ENCARG(true)) ENCARG(cons_app(cons_nonzero, filter)) -> c9(APP(nonzero, filter), ENCARG(filter)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil), ENCARG(nil)) ENCARG(cons_app(cons_nonzero, cons)) -> c9(APP(nonzero, cons), ENCARG(cons)) ENCARG(cons_app(cons_nonzero, filtersub)) -> c9(APP(nonzero, filtersub), ENCARG(filtersub)) ENCARG(cons_app(cons_nonzero, cons_app(z0, z1))) -> c9(APP(nonzero, app(encArg(z0), encArg(z1))), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero), ENCARG(cons_nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, neq)) -> c9(APP(nonzero, neq), ENCARG(neq)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0), ENCARG(0)) ENCARG(cons_app(cons_nonzero, false)) -> c9(APP(nonzero, false), ENCARG(false)) ENCARG(cons_app(cons_nonzero, s)) -> c9(APP(nonzero, s), ENCARG(s)) ENCARG(cons_app(cons_nonzero, true)) -> c9(APP(nonzero, true), ENCARG(true)) ENCARG(cons_app(cons_nonzero, filter)) -> c9(APP(nonzero, filter), ENCARG(filter)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil), ENCARG(nil)) ENCARG(cons_app(cons_nonzero, cons)) -> c9(APP(nonzero, cons), ENCARG(cons)) ENCARG(cons_app(cons_nonzero, filtersub)) -> c9(APP(nonzero, filtersub), ENCARG(filtersub)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero), ENCARG(cons_nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(cons_nonzero, cons)) -> c9(APP(nonzero, cons), ENCARG(cons)) ENCARG(cons_app(cons_nonzero, s)) -> c9(APP(nonzero, s), ENCARG(s)) ENCARG(cons_app(cons_nonzero, false)) -> c9(APP(nonzero, false), ENCARG(false)) ENCARG(cons_app(cons_nonzero, filter)) -> c9(APP(nonzero, filter), ENCARG(filter)) ENCARG(cons_app(cons_nonzero, true)) -> c9(APP(nonzero, true), ENCARG(true)) ENCARG(cons_app(cons_nonzero, neq)) -> c9(APP(nonzero, neq), ENCARG(neq)) ENCARG(cons_app(cons_nonzero, filtersub)) -> c9(APP(nonzero, filtersub), ENCARG(filtersub)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0), ENCARG(0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil), ENCARG(nil)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero), ENCARG(cons_nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2)), APP(app(neq, app(s, z0)), app(s, z1))) by APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(filter, z0), app(app(cons, z1), z2))) by APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(cons, z1), app(app(filter, z0), z2))), app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, nil), app(filter, z0)), app(app(cons, nil), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(cons, z1), app(app(filter, z0), z2))), app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, nil), app(filter, z0)), app(app(cons, nil), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(cons, z1), app(app(filter, z0), z2))), app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, nil), app(filter, z0)), app(app(cons, nil), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2)), APP(app(app(filtersub, false), z0), app(app(cons, z1), z2))) by APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, nil), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, nil), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, nil), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, 0) -> c(APP(encArg(x0), 0)) by ENCODE_APP(neq, 0) -> c(APP(neq, 0)) ENCODE_APP(0, 0) -> c(APP(0, 0)) ENCODE_APP(false, 0) -> c(APP(false, 0)) ENCODE_APP(s, 0) -> c(APP(s, 0)) ENCODE_APP(true, 0) -> c(APP(true, 0)) ENCODE_APP(filter, 0) -> c(APP(filter, 0)) ENCODE_APP(nil, 0) -> c(APP(nil, 0)) ENCODE_APP(cons, 0) -> c(APP(cons, 0)) ENCODE_APP(filtersub, 0) -> c(APP(filtersub, 0)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(neq, 0) -> c(APP(neq, 0)) ENCODE_APP(0, 0) -> c(APP(0, 0)) ENCODE_APP(false, 0) -> c(APP(false, 0)) ENCODE_APP(s, 0) -> c(APP(s, 0)) ENCODE_APP(true, 0) -> c(APP(true, 0)) ENCODE_APP(filter, 0) -> c(APP(filter, 0)) ENCODE_APP(nil, 0) -> c(APP(nil, 0)) ENCODE_APP(cons, 0) -> c(APP(cons, 0)) ENCODE_APP(filtersub, 0) -> c(APP(filtersub, 0)) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(true, 0) -> c(APP(true, 0)) ENCODE_APP(false, 0) -> c(APP(false, 0)) ENCODE_APP(filtersub, 0) -> c(APP(filtersub, 0)) ENCODE_APP(nil, 0) -> c(APP(nil, 0)) ENCODE_APP(filter, 0) -> c(APP(filter, 0)) ENCODE_APP(0, 0) -> c(APP(0, 0)) ENCODE_APP(cons, 0) -> c(APP(cons, 0)) ENCODE_APP(s, 0) -> c(APP(s, 0)) ENCODE_APP(neq, 0) -> c(APP(neq, 0)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) by ENCODE_APP(neq, nil) -> c(APP(neq, nil)) ENCODE_APP(0, nil) -> c(APP(0, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(s, nil) -> c(APP(s, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filtersub, nil) -> c(APP(filtersub, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(neq, nil) -> c(APP(neq, nil)) ENCODE_APP(0, nil) -> c(APP(0, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ENCODE_APP(s, nil) -> c(APP(s, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filtersub, nil) -> c(APP(filtersub, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ENCODE_APP(0, nil) -> c(APP(0, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(filter, nil) -> c(APP(filter, nil)) ENCODE_APP(s, nil) -> c(APP(s, nil)) ENCODE_APP(neq, nil) -> c(APP(neq, nil)) ENCODE_APP(filtersub, nil) -> c(APP(filtersub, nil)) ENCODE_APP(true, nil) -> c(APP(true, nil)) ENCODE_APP(false, nil) -> c(APP(false, nil)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) by ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(neq, x2)) -> c(APP(encArg(x0), app(neq, encArg(x2)))) ENCODE_APP(x0, cons_app(0, x2)) -> c(APP(encArg(x0), app(0, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filtersub, x2)) -> c(APP(encArg(x0), app(filtersub, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(neq, cons_app(x1, x2)) -> c(APP(neq, app(encArg(x1), encArg(x2)))) ENCODE_APP(0, cons_app(x1, x2)) -> c(APP(0, app(encArg(x1), encArg(x2)))) ENCODE_APP(false, cons_app(x1, x2)) -> c(APP(false, app(encArg(x1), encArg(x2)))) ENCODE_APP(s, cons_app(x1, x2)) -> c(APP(s, app(encArg(x1), encArg(x2)))) ENCODE_APP(true, cons_app(x1, x2)) -> c(APP(true, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter, cons_app(x1, x2)) -> c(APP(filter, app(encArg(x1), encArg(x2)))) ENCODE_APP(nil, cons_app(x1, x2)) -> c(APP(nil, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons, cons_app(x1, x2)) -> c(APP(cons, app(encArg(x1), encArg(x2)))) ENCODE_APP(filtersub, cons_app(x1, x2)) -> c(APP(filtersub, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(neq, x2)) -> c(APP(encArg(x0), app(neq, encArg(x2)))) ENCODE_APP(x0, cons_app(0, x2)) -> c(APP(encArg(x0), app(0, encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(x0, cons_app(filtersub, x2)) -> c(APP(encArg(x0), app(filtersub, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(neq, cons_app(x1, x2)) -> c(APP(neq, app(encArg(x1), encArg(x2)))) ENCODE_APP(0, cons_app(x1, x2)) -> c(APP(0, app(encArg(x1), encArg(x2)))) ENCODE_APP(false, cons_app(x1, x2)) -> c(APP(false, app(encArg(x1), encArg(x2)))) ENCODE_APP(s, cons_app(x1, x2)) -> c(APP(s, app(encArg(x1), encArg(x2)))) ENCODE_APP(true, cons_app(x1, x2)) -> c(APP(true, app(encArg(x1), encArg(x2)))) ENCODE_APP(filter, cons_app(x1, x2)) -> c(APP(filter, app(encArg(x1), encArg(x2)))) ENCODE_APP(nil, cons_app(x1, x2)) -> c(APP(nil, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons, cons_app(x1, x2)) -> c(APP(cons, app(encArg(x1), encArg(x2)))) ENCODE_APP(filtersub, cons_app(x1, x2)) -> c(APP(filtersub, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 17 trailing nodes: ENCODE_APP(filter, cons_app(x1, x2)) -> c(APP(filter, app(encArg(x1), encArg(x2)))) ENCODE_APP(x0, cons_app(filter, x2)) -> c(APP(encArg(x0), app(filter, encArg(x2)))) ENCODE_APP(x0, cons_app(nil, x2)) -> c(APP(encArg(x0), app(nil, encArg(x2)))) ENCODE_APP(s, cons_app(x1, x2)) -> c(APP(s, app(encArg(x1), encArg(x2)))) ENCODE_APP(false, cons_app(x1, x2)) -> c(APP(false, app(encArg(x1), encArg(x2)))) ENCODE_APP(filtersub, cons_app(x1, x2)) -> c(APP(filtersub, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons, cons_app(x1, x2)) -> c(APP(cons, app(encArg(x1), encArg(x2)))) ENCODE_APP(true, cons_app(x1, x2)) -> c(APP(true, app(encArg(x1), encArg(x2)))) ENCODE_APP(x0, cons_app(0, x2)) -> c(APP(encArg(x0), app(0, encArg(x2)))) ENCODE_APP(neq, cons_app(x1, x2)) -> c(APP(neq, app(encArg(x1), encArg(x2)))) ENCODE_APP(x0, cons_app(false, x2)) -> c(APP(encArg(x0), app(false, encArg(x2)))) ENCODE_APP(x0, cons_app(filtersub, x2)) -> c(APP(encArg(x0), app(filtersub, encArg(x2)))) ENCODE_APP(x0, cons_app(cons, x2)) -> c(APP(encArg(x0), app(cons, encArg(x2)))) ENCODE_APP(nil, cons_app(x1, x2)) -> c(APP(nil, app(encArg(x1), encArg(x2)))) ENCODE_APP(x0, cons_app(true, x2)) -> c(APP(encArg(x0), app(true, encArg(x2)))) ENCODE_APP(x0, cons_app(neq, x2)) -> c(APP(encArg(x0), app(neq, encArg(x2)))) ENCODE_APP(0, cons_app(x1, x2)) -> c(APP(0, app(encArg(x1), encArg(x2)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), nonzero)) by ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), app(filter, app(neq, 0)))) ENCODE_APP(neq, cons_nonzero) -> c(APP(neq, nonzero)) ENCODE_APP(0, cons_nonzero) -> c(APP(0, nonzero)) ENCODE_APP(false, cons_nonzero) -> c(APP(false, nonzero)) ENCODE_APP(s, cons_nonzero) -> c(APP(s, nonzero)) ENCODE_APP(true, cons_nonzero) -> c(APP(true, nonzero)) ENCODE_APP(filter, cons_nonzero) -> c(APP(filter, nonzero)) ENCODE_APP(nil, cons_nonzero) -> c(APP(nil, nonzero)) ENCODE_APP(cons, cons_nonzero) -> c(APP(cons, nonzero)) ENCODE_APP(filtersub, cons_nonzero) -> c(APP(filtersub, nonzero)) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), app(filter, app(neq, 0)))) ENCODE_APP(neq, cons_nonzero) -> c(APP(neq, nonzero)) ENCODE_APP(0, cons_nonzero) -> c(APP(0, nonzero)) ENCODE_APP(false, cons_nonzero) -> c(APP(false, nonzero)) ENCODE_APP(s, cons_nonzero) -> c(APP(s, nonzero)) ENCODE_APP(true, cons_nonzero) -> c(APP(true, nonzero)) ENCODE_APP(filter, cons_nonzero) -> c(APP(filter, nonzero)) ENCODE_APP(nil, cons_nonzero) -> c(APP(nil, nonzero)) ENCODE_APP(cons, cons_nonzero) -> c(APP(cons, nonzero)) ENCODE_APP(filtersub, cons_nonzero) -> c(APP(filtersub, nonzero)) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: ENCODE_APP(filtersub, cons_nonzero) -> c(APP(filtersub, nonzero)) ENCODE_APP(s, cons_nonzero) -> c(APP(s, nonzero)) ENCODE_APP(cons, cons_nonzero) -> c(APP(cons, nonzero)) ENCODE_APP(nil, cons_nonzero) -> c(APP(nil, nonzero)) ENCODE_APP(0, cons_nonzero) -> c(APP(0, nonzero)) ENCODE_APP(false, cons_nonzero) -> c(APP(false, nonzero)) ENCODE_APP(neq, cons_nonzero) -> c(APP(neq, nonzero)) ENCODE_APP(x0, cons_nonzero) -> c(APP(encArg(x0), app(filter, app(neq, 0)))) ENCODE_APP(filter, cons_nonzero) -> c(APP(filter, nonzero)) ENCODE_APP(true, cons_nonzero) -> c(APP(true, nonzero)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) by ENCODE_APP(cons_app(x0, x1), neq) -> c(APP(app(encArg(x0), encArg(x1)), neq)) ENCODE_APP(cons_app(x0, x1), 0) -> c(APP(app(encArg(x0), encArg(x1)), 0)) ENCODE_APP(cons_app(x0, x1), false) -> c(APP(app(encArg(x0), encArg(x1)), false)) ENCODE_APP(cons_app(x0, x1), s) -> c(APP(app(encArg(x0), encArg(x1)), s)) ENCODE_APP(cons_app(x0, x1), true) -> c(APP(app(encArg(x0), encArg(x1)), true)) ENCODE_APP(cons_app(x0, x1), filter) -> c(APP(app(encArg(x0), encArg(x1)), filter)) ENCODE_APP(cons_app(x0, x1), nil) -> c(APP(app(encArg(x0), encArg(x1)), nil)) ENCODE_APP(cons_app(x0, x1), cons) -> c(APP(app(encArg(x0), encArg(x1)), cons)) ENCODE_APP(cons_app(x0, x1), filtersub) -> c(APP(app(encArg(x0), encArg(x1)), filtersub)) ENCODE_APP(cons_app(x0, x1), cons_app(z0, z1)) -> c(APP(app(encArg(x0), encArg(x1)), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(x0, x1), cons_nonzero) -> c(APP(app(encArg(x0), encArg(x1)), nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(0, x1), x2) -> c(APP(app(0, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(false, x1), x2) -> c(APP(app(false, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(s, x1), x2) -> c(APP(app(s, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(true, x1), x2) -> c(APP(app(true, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(nil, x1), x2) -> c(APP(app(nil, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons, x1), x2) -> c(APP(app(cons, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filtersub, x1), x2) -> c(APP(app(filtersub, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, x1), neq) -> c(APP(app(encArg(x0), encArg(x1)), neq)) ENCODE_APP(cons_app(x0, x1), false) -> c(APP(app(encArg(x0), encArg(x1)), false)) ENCODE_APP(cons_app(x0, x1), s) -> c(APP(app(encArg(x0), encArg(x1)), s)) ENCODE_APP(cons_app(x0, x1), true) -> c(APP(app(encArg(x0), encArg(x1)), true)) ENCODE_APP(cons_app(x0, x1), filter) -> c(APP(app(encArg(x0), encArg(x1)), filter)) ENCODE_APP(cons_app(x0, x1), cons) -> c(APP(app(encArg(x0), encArg(x1)), cons)) ENCODE_APP(cons_app(x0, x1), filtersub) -> c(APP(app(encArg(x0), encArg(x1)), filtersub)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(0, x1), x2) -> c(APP(app(0, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(false, x1), x2) -> c(APP(app(false, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(s, x1), x2) -> c(APP(app(s, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(true, x1), x2) -> c(APP(app(true, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(nil, x1), x2) -> c(APP(app(nil, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons, x1), x2) -> c(APP(app(cons, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filtersub, x1), x2) -> c(APP(app(filtersub, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 14 trailing nodes: ENCODE_APP(cons_app(x0, x1), true) -> c(APP(app(encArg(x0), encArg(x1)), true)) ENCODE_APP(cons_app(cons, x1), x2) -> c(APP(app(cons, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(x0, x1), cons) -> c(APP(app(encArg(x0), encArg(x1)), cons)) ENCODE_APP(cons_app(x0, x1), s) -> c(APP(app(encArg(x0), encArg(x1)), s)) ENCODE_APP(cons_app(filtersub, x1), x2) -> c(APP(app(filtersub, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(x0, x1), filter) -> c(APP(app(encArg(x0), encArg(x1)), filter)) ENCODE_APP(cons_app(x0, x1), filtersub) -> c(APP(app(encArg(x0), encArg(x1)), filtersub)) ENCODE_APP(cons_app(x0, x1), neq) -> c(APP(app(encArg(x0), encArg(x1)), neq)) ENCODE_APP(cons_app(s, x1), x2) -> c(APP(app(s, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(false, x1), x2) -> c(APP(app(false, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(0, x1), x2) -> c(APP(app(0, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(x0, x1), false) -> c(APP(app(encArg(x0), encArg(x1)), false)) ENCODE_APP(cons_app(true, x1), x2) -> c(APP(app(true, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(nil, x1), x2) -> c(APP(app(nil, encArg(x1)), encArg(x2))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c_1, c9_3, c9_2 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(cons_nonzero, x1) -> c(APP(nonzero, encArg(x1))) by ENCODE_APP(cons_nonzero, neq) -> c(APP(nonzero, neq)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_nonzero, false) -> c(APP(nonzero, false)) ENCODE_APP(cons_nonzero, s) -> c(APP(nonzero, s)) ENCODE_APP(cons_nonzero, true) -> c(APP(nonzero, true)) ENCODE_APP(cons_nonzero, filter) -> c(APP(nonzero, filter)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(cons_nonzero, cons) -> c(APP(nonzero, cons)) ENCODE_APP(cons_nonzero, filtersub) -> c(APP(nonzero, filtersub)) ENCODE_APP(cons_nonzero, cons_app(z0, z1)) -> c(APP(nonzero, app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_nonzero, x0) -> c(APP(app(filter, app(neq, 0)), encArg(x0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) ENCODE_APP(cons_nonzero, neq) -> c(APP(nonzero, neq)) ENCODE_APP(cons_nonzero, false) -> c(APP(nonzero, false)) ENCODE_APP(cons_nonzero, s) -> c(APP(nonzero, s)) ENCODE_APP(cons_nonzero, true) -> c(APP(nonzero, true)) ENCODE_APP(cons_nonzero, filter) -> c(APP(nonzero, filter)) ENCODE_APP(cons_nonzero, cons) -> c(APP(nonzero, cons)) ENCODE_APP(cons_nonzero, filtersub) -> c(APP(nonzero, filtersub)) ENCODE_APP(cons_nonzero, x0) -> c(APP(app(filter, app(neq, 0)), encArg(x0))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c9_3, c9_2, c_1 ---------------------------------------- (127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(cons_nonzero, filtersub) -> c(APP(nonzero, filtersub)) ENCODE_APP(cons_nonzero, cons) -> c(APP(nonzero, cons)) ENCODE_APP(cons_nonzero, s) -> c(APP(nonzero, s)) ENCODE_APP(cons_nonzero, false) -> c(APP(nonzero, false)) ENCODE_APP(cons_nonzero, filter) -> c(APP(nonzero, filter)) ENCODE_APP(cons_nonzero, true) -> c(APP(nonzero, true)) ENCODE_APP(cons_nonzero, neq) -> c(APP(nonzero, neq)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) ENCODE_APP(cons_nonzero, x0) -> c(APP(app(filter, app(neq, 0)), encArg(x0))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c25_1, c28_1, c29_1, c9_1, c27_2, c27_1, c9_3, c9_2, c_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(app(neq, app(s, z0)), app(s, z1)) -> c25(APP(app(neq, z0), z1)) by APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) ENCODE_APP(cons_nonzero, x0) -> c(APP(app(filter, app(neq, 0)), encArg(x0))) APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c28_1, c29_1, c9_1, c27_2, c27_1, c9_3, c9_2, c_1, c25_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> c28(APP(app(filter, z0), z2)) by APP(app(app(filtersub, true), z0), app(app(cons, z1), nil)) -> c28(APP(app(filter, z0), nil)) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, 0), y0))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, 0), y0))) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, app(s, y0)), y1))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, app(s, y0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, 0), y1))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, 0), y1))) APP(app(app(filtersub, true), app(filter, y0)), app(app(cons, z1), app(app(cons, nil), y1))) -> c28(APP(app(filter, app(filter, y0)), app(app(cons, nil), y1))) APP(app(app(filtersub, true), app(app(filtersub, true), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, true), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, 0)), y0))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), y0))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, app(s, y0))), y1))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, y0))), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, 0)), y1))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, 0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, app(s, y1))), y2))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, app(s, y1))), y2))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, app(s, y1)), y2))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, app(s, y1)), y2))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, false), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, false), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y0)), y1))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), y0)), y1))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, 0), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, true), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, true), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, nil), y1)), y2))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, nil), y1)), y2))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), nil)), y2))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), nil)), y2))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) ENCODE_APP(cons_nonzero, x0) -> c(APP(app(filter, app(neq, 0)), encArg(x0))) APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) APP(app(app(filtersub, true), z0), app(app(cons, z1), nil)) -> c28(APP(app(filter, z0), nil)) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, 0), y0))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, 0), y0))) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, app(s, y0)), y1))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, app(s, y0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, 0), y1))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, 0), y1))) APP(app(app(filtersub, true), app(filter, y0)), app(app(cons, z1), app(app(cons, nil), y1))) -> c28(APP(app(filter, app(filter, y0)), app(app(cons, nil), y1))) APP(app(app(filtersub, true), app(app(filtersub, true), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, true), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, 0)), y0))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), y0))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, app(s, y0))), y1))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, y0))), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, 0)), y1))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, 0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, app(s, y1))), y2))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, app(s, y1))), y2))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, app(s, y1)), y2))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, app(s, y1)), y2))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, false), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, false), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y0)), y1))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), y0)), y1))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, 0), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, true), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, true), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, nil), y1)), y2))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, nil), y1)), y2))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), nil)), y2))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), nil)), y2))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) APP(app(app(filtersub, true), z0), app(app(cons, z1), nil)) -> c28(APP(app(filter, z0), nil)) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, 0), y0))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, 0), y0))) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, app(s, y0)), y1))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, app(s, y0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, 0), y1))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, 0), y1))) APP(app(app(filtersub, true), app(filter, y0)), app(app(cons, z1), app(app(cons, nil), y1))) -> c28(APP(app(filter, app(filter, y0)), app(app(cons, nil), y1))) APP(app(app(filtersub, true), app(app(filtersub, true), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, true), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, 0)), y0))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), y0))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, app(s, y0))), y1))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, y0))), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, 0)), y1))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, 0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, app(s, y1))), y2))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, app(s, y1))), y2))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, app(s, y1)), y2))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, app(s, y1)), y2))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, false), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, false), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y0)), y1))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), y0)), y1))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, 0), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, true), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, true), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, nil), y1)), y2))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, nil), y1)), y2))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), nil)), y2))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), nil)), y2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c29_1, c9_1, c27_2, c27_1, c9_3, c9_2, c_1, c25_1, c28_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace ENCARG(cons_app(x0, cons_app(x1, cons_nonzero))) -> c9(APP(encArg(x0), app(encArg(x1), nonzero)), ENCARG(x0), ENCARG(cons_app(x1, cons_nonzero))) by ENCARG(cons_app(z0, cons_app(z1, cons_nonzero))) -> c9(APP(encArg(z0), app(encArg(z1), app(filter, app(neq, 0)))), ENCARG(z0), ENCARG(cons_app(z1, cons_nonzero))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(neq) -> neq encArg(0) -> 0 encArg(false) -> false encArg(s) -> s encArg(true) -> true encArg(filter) -> filter encArg(nil) -> nil encArg(cons) -> cons encArg(filtersub) -> filtersub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encArg(cons_nonzero) -> nonzero app(app(neq, 0), 0) -> false app(app(neq, 0), app(s, z0)) -> true app(app(neq, app(s, z0)), 0) -> true app(app(neq, app(s, z0)), app(s, z1)) -> app(app(neq, z0), z1) app(app(filter, z0), nil) -> nil app(app(filter, z0), app(app(cons, z1), z2)) -> app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2)) app(app(app(filtersub, true), z0), app(app(cons, z1), z2)) -> app(app(cons, z1), app(app(filter, z0), z2)) app(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> app(app(filter, z0), z2) nonzero -> app(filter, app(neq, 0)) Tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) ENCARG(cons_app(x0, neq)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, false)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, s)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, true)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filter)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c9(ENCARG(x0)) ENCARG(cons_app(x0, filtersub)) -> c9(ENCARG(x0)) ENCARG(cons_app(neq, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(0, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(false, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(s, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(true, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filter, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c9(ENCARG(x1)) ENCARG(cons_app(filtersub, x1)) -> c9(ENCARG(x1)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) ENCARG(cons_app(x0, cons_app(x1, neq))) -> c9(APP(encArg(x0), app(encArg(x1), neq)), ENCARG(x0), ENCARG(cons_app(x1, neq))) ENCARG(cons_app(x0, cons_app(x1, 0))) -> c9(APP(encArg(x0), app(encArg(x1), 0)), ENCARG(x0), ENCARG(cons_app(x1, 0))) ENCARG(cons_app(x0, cons_app(x1, false))) -> c9(APP(encArg(x0), app(encArg(x1), false)), ENCARG(x0), ENCARG(cons_app(x1, false))) ENCARG(cons_app(x0, cons_app(x1, s))) -> c9(APP(encArg(x0), app(encArg(x1), s)), ENCARG(x0), ENCARG(cons_app(x1, s))) ENCARG(cons_app(x0, cons_app(x1, true))) -> c9(APP(encArg(x0), app(encArg(x1), true)), ENCARG(x0), ENCARG(cons_app(x1, true))) ENCARG(cons_app(x0, cons_app(x1, filter))) -> c9(APP(encArg(x0), app(encArg(x1), filter)), ENCARG(x0), ENCARG(cons_app(x1, filter))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c9(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c9(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, filtersub))) -> c9(APP(encArg(x0), app(encArg(x1), filtersub)), ENCARG(x0), ENCARG(cons_app(x1, filtersub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c9(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_app(x1, cons_app(z0, z1)))) ENCARG(cons_app(x0, cons_app(s, x2))) -> c9(APP(encArg(x0), app(s, encArg(x2))), ENCARG(x0), ENCARG(cons_app(s, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c9(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(x0, cons_app(cons_nonzero, x2))) -> c9(APP(encArg(x0), app(nonzero, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_nonzero, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c9(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(x0, cons_app(neq, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(neq, x2))) ENCARG(cons_app(x0, cons_app(0, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(0, x2))) ENCARG(cons_app(x0, cons_app(false, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(false, x2))) ENCARG(cons_app(x0, cons_app(true, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(true, x2))) ENCARG(cons_app(x0, cons_app(filter, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filter, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(filtersub, x2))) -> c9(ENCARG(x0), ENCARG(cons_app(filtersub, x2))) ENCARG(cons_app(neq, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(0, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(false, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(s, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(true, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filter, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(filtersub, cons_app(x1, x2))) -> c9(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_nonzero, cons_app(x1, x2))) -> c9(APP(nonzero, app(encArg(x1), encArg(x2))), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, neq), x2)) -> c9(APP(app(encArg(x0), neq), encArg(x2)), ENCARG(cons_app(x0, neq)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, 0), x2)) -> c9(APP(app(encArg(x0), 0), encArg(x2)), ENCARG(cons_app(x0, 0)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, false), x2)) -> c9(APP(app(encArg(x0), false), encArg(x2)), ENCARG(cons_app(x0, false)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, s), x2)) -> c9(APP(app(encArg(x0), s), encArg(x2)), ENCARG(cons_app(x0, s)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, true), x2)) -> c9(APP(app(encArg(x0), true), encArg(x2)), ENCARG(cons_app(x0, true)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filter), x2)) -> c9(APP(app(encArg(x0), filter), encArg(x2)), ENCARG(cons_app(x0, filter)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c9(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c9(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, filtersub), x2)) -> c9(APP(app(encArg(x0), filtersub), encArg(x2)), ENCARG(cons_app(x0, filtersub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c9(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2)), ENCARG(cons_app(x0, cons_app(z0, z1))), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_nonzero), x2)) -> c9(APP(app(encArg(x0), nonzero), encArg(x2)), ENCARG(cons_app(x0, cons_nonzero)), ENCARG(x2)) ENCARG(cons_app(cons_app(neq, x1), x2)) -> c9(APP(app(neq, encArg(x1)), encArg(x2)), ENCARG(cons_app(neq, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filter, x1), x2)) -> c9(APP(app(filter, encArg(x1)), encArg(x2)), ENCARG(cons_app(filter, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c9(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_nonzero, x1), x2)) -> c9(APP(app(nonzero, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_nonzero, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), neq)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), 0)) -> c9(APP(app(encArg(x0), encArg(x1)), 0), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), false)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), s)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), true)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filter)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c9(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), filtersub)) -> c9(ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons_nonzero)) -> c9(APP(app(encArg(x0), encArg(x1)), nonzero), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(0, x1), x2)) -> c9(ENCARG(cons_app(0, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(false, x1), x2)) -> c9(ENCARG(cons_app(false, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(s, x1), x2)) -> c9(ENCARG(cons_app(s, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(true, x1), x2)) -> c9(ENCARG(cons_app(true, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c9(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c9(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(filtersub, x1), x2)) -> c9(ENCARG(cons_app(filtersub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_nonzero, 0)) -> c9(APP(nonzero, 0)) ENCARG(cons_app(cons_nonzero, nil)) -> c9(APP(nonzero, nil)) ENCARG(cons_app(x0, cons_nonzero)) -> c9(ENCARG(x0)) ENCARG(cons_app(cons_nonzero, cons_nonzero)) -> c9(APP(nonzero, nonzero)) ENCARG(cons_app(cons_nonzero, x0)) -> c9(APP(app(filter, app(neq, 0)), encArg(x0)), ENCARG(x0)) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) ENCODE_APP(cons_app(z0, z1), 0) -> c(APP(app(encArg(z0), encArg(z1)), 0)) ENCODE_APP(cons_nonzero, 0) -> c(APP(nonzero, 0)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_nonzero, nil) -> c(APP(nonzero, nil)) ENCODE_APP(x0, cons_app(x1, neq)) -> c(APP(encArg(x0), app(encArg(x1), neq))) ENCODE_APP(x0, cons_app(x1, 0)) -> c(APP(encArg(x0), app(encArg(x1), 0))) ENCODE_APP(x0, cons_app(x1, false)) -> c(APP(encArg(x0), app(encArg(x1), false))) ENCODE_APP(x0, cons_app(x1, s)) -> c(APP(encArg(x0), app(encArg(x1), s))) ENCODE_APP(x0, cons_app(x1, true)) -> c(APP(encArg(x0), app(encArg(x1), true))) ENCODE_APP(x0, cons_app(x1, filter)) -> c(APP(encArg(x0), app(encArg(x1), filter))) ENCODE_APP(x0, cons_app(x1, nil)) -> c(APP(encArg(x0), app(encArg(x1), nil))) ENCODE_APP(x0, cons_app(x1, cons)) -> c(APP(encArg(x0), app(encArg(x1), cons))) ENCODE_APP(x0, cons_app(x1, filtersub)) -> c(APP(encArg(x0), app(encArg(x1), filtersub))) ENCODE_APP(x0, cons_app(x1, cons_app(z0, z1))) -> c(APP(encArg(x0), app(encArg(x1), app(encArg(z0), encArg(z1))))) ENCODE_APP(x0, cons_app(x1, cons_nonzero)) -> c(APP(encArg(x0), app(encArg(x1), nonzero))) ENCODE_APP(x0, cons_app(s, x2)) -> c(APP(encArg(x0), app(s, encArg(x2)))) ENCODE_APP(x0, cons_app(cons_app(z0, z1), x2)) -> c(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_APP(x0, cons_app(cons_nonzero, x2)) -> c(APP(encArg(x0), app(nonzero, encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_nonzero, cons_app(x1, x2)) -> c(APP(nonzero, app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(z0, z1), cons_nonzero) -> c(APP(app(encArg(z0), encArg(z1)), nonzero)) ENCODE_APP(cons_nonzero, cons_nonzero) -> c(APP(nonzero, nonzero)) ENCODE_APP(cons_app(x0, neq), x2) -> c(APP(app(encArg(x0), neq), encArg(x2))) ENCODE_APP(cons_app(x0, 0), x2) -> c(APP(app(encArg(x0), 0), encArg(x2))) ENCODE_APP(cons_app(x0, false), x2) -> c(APP(app(encArg(x0), false), encArg(x2))) ENCODE_APP(cons_app(x0, s), x2) -> c(APP(app(encArg(x0), s), encArg(x2))) ENCODE_APP(cons_app(x0, true), x2) -> c(APP(app(encArg(x0), true), encArg(x2))) ENCODE_APP(cons_app(x0, filter), x2) -> c(APP(app(encArg(x0), filter), encArg(x2))) ENCODE_APP(cons_app(x0, nil), x2) -> c(APP(app(encArg(x0), nil), encArg(x2))) ENCODE_APP(cons_app(x0, cons), x2) -> c(APP(app(encArg(x0), cons), encArg(x2))) ENCODE_APP(cons_app(x0, filtersub), x2) -> c(APP(app(encArg(x0), filtersub), encArg(x2))) ENCODE_APP(cons_app(x0, cons_app(z0, z1)), x2) -> c(APP(app(encArg(x0), app(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_APP(cons_app(x0, cons_nonzero), x2) -> c(APP(app(encArg(x0), nonzero), encArg(x2))) ENCODE_APP(cons_app(neq, x1), x2) -> c(APP(app(neq, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(filter, x1), x2) -> c(APP(app(filter, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_app(z0, z1), x1), x2) -> c(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(cons_nonzero, x1), x2) -> c(APP(app(nonzero, encArg(x1)), encArg(x2))) ENCODE_APP(cons_nonzero, x0) -> c(APP(app(filter, app(neq, 0)), encArg(x0))) APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) APP(app(app(filtersub, true), z0), app(app(cons, z1), nil)) -> c28(APP(app(filter, z0), nil)) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, 0), y0))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, 0), y0))) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, app(s, y0)), y1))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, app(s, y0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, 0), y1))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, 0), y1))) APP(app(app(filtersub, true), app(filter, y0)), app(app(cons, z1), app(app(cons, nil), y1))) -> c28(APP(app(filter, app(filter, y0)), app(app(cons, nil), y1))) APP(app(app(filtersub, true), app(app(filtersub, true), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, true), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, 0)), y0))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), y0))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, app(s, y0))), y1))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, y0))), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, 0)), y1))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, 0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, app(s, y1))), y2))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, app(s, y1))), y2))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, app(s, y1)), y2))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, app(s, y1)), y2))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, false), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, false), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y0)), y1))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), y0)), y1))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, 0), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, true), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, true), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, nil), y1)), y2))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, nil), y1)), y2))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), nil)), y2))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), nil)), y2))) ENCARG(cons_app(z0, cons_app(z1, cons_nonzero))) -> c9(APP(encArg(z0), app(encArg(z1), app(filter, app(neq, 0)))), ENCARG(z0), ENCARG(cons_app(z1, cons_nonzero))) S tuples: APP(app(neq, 0), 0) -> c22 APP(app(neq, 0), app(s, z0)) -> c23 APP(app(neq, app(s, z0)), 0) -> c24 APP(app(filter, z0), nil) -> c26 APP(app(app(filtersub, false), z0), app(app(cons, z1), z2)) -> c29(APP(app(filter, z0), z2)) APP(app(filter, app(neq, 0)), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2)), APP(app(neq, 0), 0)) APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2)), APP(app(neq, 0), app(s, z0))) APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2)), APP(app(neq, app(s, z0)), 0)) APP(app(filter, app(filter, z0)), app(app(cons, nil), x2)) -> c27(APP(app(filter, z0), nil)) APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2)) -> c27(APP(app(app(filtersub, true), z0), app(app(cons, z1), z2))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, false), app(neq, app(s, 0))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, 0)), app(s, 0))) APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, app(s, app(s, z0))), x2)), APP(app(neq, app(s, 0)), app(s, app(s, z0)))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)) -> c27(APP(app(app(filtersub, true), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, 0)), x2)), APP(app(neq, app(s, app(s, z0))), app(s, 0))) APP(app(filter, app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)) -> c27(APP(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, app(s, z0)))), app(app(cons, app(s, app(s, z1))), x2)), APP(app(neq, app(s, app(s, z0))), app(s, app(s, z1)))) APP(app(filter, app(neq, app(s, x0))), app(app(cons, app(s, x1)), x2)) -> c27(APP(app(neq, app(s, x0)), app(s, x1))) APP(app(filter, app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(filter, z0), z2)), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(app(filtersub, false), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, false), app(neq, 0)), app(app(cons, 0), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, 0)), app(app(cons, app(s, z0)), x2))), app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, z0)), x2)), x3)), APP(app(filter, app(neq, 0)), app(app(cons, app(s, z0)), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, true), app(neq, app(s, z0))), app(app(cons, 0), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, 0), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, 0), x2))) APP(app(filter, app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(neq, z0), z1)), app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))), app(filter, app(neq, app(s, z0)))), app(app(cons, app(app(cons, app(s, z1)), x2)), x3)), APP(app(filter, app(neq, app(s, z0))), app(app(cons, app(s, z1)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))), app(filter, app(filter, z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)), APP(app(filter, app(filter, z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(app(filtersub, true), z0))), app(app(cons, app(app(cons, app(app(cons, z1), z2)), x2)), x3)) -> c27(APP(app(filter, app(app(filtersub, true), z0)), app(app(cons, app(app(cons, z1), z2)), x2))) APP(app(filter, app(filter, app(filter, z0))), app(app(cons, app(app(cons, nil), x2)), x3)) -> c27(APP(app(filter, app(filter, z0)), app(app(cons, nil), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)) -> c27(APP(app(app(filtersub, app(app(app(filtersub, app(z0, z1)), z0), app(app(cons, z1), z2))), app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), app(app(cons, z1), z2))), x3)), APP(app(app(filtersub, false), z0), app(app(cons, x1), app(app(cons, z1), z2)))) APP(app(filter, app(app(filtersub, false), x0)), app(app(cons, app(app(cons, x1), x2)), x3)) -> c27(APP(app(app(filtersub, false), x0), app(app(cons, x1), x2))) APP(app(filter, app(app(filtersub, false), z0)), app(app(cons, app(app(cons, x1), nil)), x3)) -> c27(APP(app(app(filtersub, false), z0), app(app(cons, x1), nil))) APP(app(neq, app(s, 0)), app(s, 0)) -> c25(APP(app(neq, 0), 0)) APP(app(neq, app(s, 0)), app(s, app(s, y0))) -> c25(APP(app(neq, 0), app(s, y0))) APP(app(neq, app(s, app(s, y0))), app(s, 0)) -> c25(APP(app(neq, app(s, y0)), 0)) APP(app(neq, app(s, app(s, y0))), app(s, app(s, y1))) -> c25(APP(app(neq, app(s, y0)), app(s, y1))) APP(app(app(filtersub, true), z0), app(app(cons, z1), nil)) -> c28(APP(app(filter, z0), nil)) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, 0), y0))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, 0), y0))) APP(app(app(filtersub, true), app(neq, 0)), app(app(cons, z1), app(app(cons, app(s, y0)), y1))) -> c28(APP(app(filter, app(neq, 0)), app(app(cons, app(s, y0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, 0), y1))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, 0), y1))) APP(app(app(filtersub, true), app(filter, y0)), app(app(cons, z1), app(app(cons, nil), y1))) -> c28(APP(app(filter, app(filter, y0)), app(app(cons, nil), y1))) APP(app(app(filtersub, true), app(app(filtersub, true), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, true), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, 0)), y0))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, 0)), y0))) APP(app(app(filtersub, true), app(neq, app(s, 0))), app(app(cons, z1), app(app(cons, app(s, app(s, y0))), y1))) -> c28(APP(app(filter, app(neq, app(s, 0))), app(app(cons, app(s, app(s, y0))), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, 0)), y1))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, 0)), y1))) APP(app(app(filtersub, true), app(neq, app(s, app(s, y0)))), app(app(cons, z1), app(app(cons, app(s, app(s, y1))), y2))) -> c28(APP(app(filter, app(neq, app(s, app(s, y0)))), app(app(cons, app(s, app(s, y1))), y2))) APP(app(app(filtersub, true), app(neq, app(s, y0))), app(app(cons, z1), app(app(cons, app(s, y1)), y2))) -> c28(APP(app(filter, app(neq, app(s, y0))), app(app(cons, app(s, y1)), y2))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, false), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, false), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y0)), y1))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, 0), y0)), y1))) APP(app(app(filtersub, true), app(filter, app(neq, 0))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, 0))), app(app(cons, app(app(cons, app(s, y0)), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, 0), y1)), y2))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, 0), y1)), y2))) APP(app(app(filtersub, true), app(filter, app(neq, app(s, y0)))), app(app(cons, z1), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) -> c28(APP(app(filter, app(filter, app(neq, app(s, y0)))), app(app(cons, app(app(cons, app(s, y1)), y2)), y3))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(app(filtersub, true), y0))), app(app(cons, z1), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) -> c28(APP(app(filter, app(filter, app(app(filtersub, true), y0))), app(app(cons, app(app(cons, app(app(cons, y1), y2)), y3)), y4))) APP(app(app(filtersub, true), app(filter, app(filter, y0))), app(app(cons, z1), app(app(cons, app(app(cons, nil), y1)), y2))) -> c28(APP(app(filter, app(filter, app(filter, y0))), app(app(cons, app(app(cons, nil), y1)), y2))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), app(app(cons, y2), y3))), y4))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), y2)), y3))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), y2)), y3))) APP(app(app(filtersub, true), app(app(filtersub, false), y0)), app(app(cons, z1), app(app(cons, app(app(cons, y1), nil)), y2))) -> c28(APP(app(filter, app(app(filtersub, false), y0)), app(app(cons, app(app(cons, y1), nil)), y2))) K tuples:none Defined Rule Symbols: encArg_1, app_2, nonzero Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c22, c23, c24, c26, c29_1, c9_1, c27_2, c27_1, c9_3, c9_2, c_1, c25_1, c28_1