/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 205 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), 2946 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), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 21 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 4 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 35 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 17 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 46 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 247 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 199 ms] (148) 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(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub ---------------------------------------- (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(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) The (relative) TRS S consists of the following rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub 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(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) The (relative) TRS S consists of the following rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub 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(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) The (relative) TRS S consists of the following rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub 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(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) The (relative) TRS S consists of the following rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: app(app(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub Types: app :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app append :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app nil :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app cons :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app flatwith :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app leaf :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app node :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app flatwithsub :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encArg :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app cons_app :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_app :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_append :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_nil :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_cons :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_flatwith :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_leaf :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_node :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_flatwithsub :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app hole_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app1_0 :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0 :: Nat -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (12) Obligation: Innermost TRS: Rules: app(app(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub Types: app :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app append :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app nil :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app cons :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app flatwith :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app leaf :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app node :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app flatwithsub :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encArg :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app cons_app :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_app :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_append :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_nil :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_cons :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_flatwith :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_leaf :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_node :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app encode_flatwithsub :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app hole_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app1_0 :: append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0 :: Nat -> append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app Generator Equations: gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(0) <=> append gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(+(x, 1)) <=> cons_app(append, gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(x)) The following defined symbols remain to be analysed: app, encArg They will be analysed ascendingly in the following order: app < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(n331_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(0)) Induction Step: encArg(gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(+(n331_0, 1))) ->_R^Omega(0) app(encArg(append), encArg(gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(n331_0))) ->_R^Omega(0) app(append, encArg(gen_append:nil:cons:flatwith:leaf:node:flatwithsub:cons_app2_0(n331_0))) ->_IH app(append, *3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF) ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(app(append, nil), ys) -> ys app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(app(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil) app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs) app(app(flatwithsub, f), nil) -> nil app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) app(c_app(flatwithsub, f), nil) -> nil app(c_app(append, nil), ys) -> ys app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) app(c_app(flatwith, f), c_app(node, xs)) -> app(app(flatwithsub, f), xs) app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, app(f, x)), nil) The (relative) TRS S consists of the following rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub app(x0, x1) -> c_app(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) [1] app(c_app(flatwithsub, f), nil) -> nil [1] app(c_app(append, nil), ys) -> ys [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) [1] app(c_app(flatwith, f), c_app(node, xs)) -> app(app(flatwithsub, f), xs) [1] app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, app(f, x)), nil) [1] encArg(append) -> append [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(flatwith) -> flatwith [0] encArg(leaf) -> leaf [0] encArg(node) -> node [0] encArg(flatwithsub) -> flatwithsub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_append -> append [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_flatwith -> flatwith [0] encode_leaf -> leaf [0] encode_node -> node [0] encode_flatwithsub -> flatwithsub [0] app(x0, x1) -> c_app(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) [1] app(c_app(flatwithsub, f), nil) -> nil [1] app(c_app(append, nil), ys) -> ys [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) [1] app(c_app(flatwith, f), c_app(node, xs)) -> app(app(flatwithsub, f), xs) [1] app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, app(f, x)), nil) [1] encArg(append) -> append [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(flatwith) -> flatwith [0] encArg(leaf) -> leaf [0] encArg(node) -> node [0] encArg(flatwithsub) -> flatwithsub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_append -> append [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_flatwith -> flatwith [0] encode_leaf -> leaf [0] encode_node -> node [0] encode_flatwithsub -> flatwithsub [0] app(x0, x1) -> c_app(x0, x1) [0] The TRS has the following type information: app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app c_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app cons_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app Rewrite Strategy: INNERMOST ---------------------------------------- (23) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_app_2 encode_append encode_nil encode_cons encode_flatwith encode_leaf encode_node encode_flatwithsub app_2 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_append -> null_encode_append [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_flatwith -> null_encode_flatwith [0] encode_leaf -> null_encode_leaf [0] encode_node -> null_encode_node [0] encode_flatwithsub -> null_encode_flatwithsub [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_append, null_encode_nil, null_encode_cons, null_encode_flatwith, null_encode_leaf, null_encode_node, null_encode_flatwithsub, null_app ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) [1] app(c_app(flatwithsub, f), nil) -> nil [1] app(c_app(append, nil), ys) -> ys [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) [1] app(c_app(flatwith, f), c_app(node, xs)) -> app(app(flatwithsub, f), xs) [1] app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, app(f, x)), nil) [1] encArg(append) -> append [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(flatwith) -> flatwith [0] encArg(leaf) -> leaf [0] encArg(node) -> node [0] encArg(flatwithsub) -> flatwithsub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_append -> append [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_flatwith -> flatwith [0] encode_leaf -> leaf [0] encode_node -> node [0] encode_flatwithsub -> flatwithsub [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_append -> null_encode_append [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_flatwith -> null_encode_flatwith [0] encode_leaf -> null_encode_leaf [0] encode_node -> null_encode_node [0] encode_flatwithsub -> null_encode_flatwithsub [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app c_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app cons_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(c_app(cons, x), app(c_app(append, xs), ys)) [1] app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(c_app(cons, x), app(null_app, ys)) [1] app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(null_app, app(c_app(append, xs), ys)) [1] app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(null_app, app(null_app, ys)) [1] app(c_app(flatwithsub, f), nil) -> nil [1] app(c_app(append, nil), ys) -> ys [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(c_app(flatwith, f), x)), app(c_app(flatwithsub, f), xs)) [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(c_app(flatwith, f), x)), app(null_app, xs)) [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(null_app, x)), app(c_app(flatwithsub, f), xs)) [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(null_app, x)), app(null_app, xs)) [1] app(c_app(flatwith, f), c_app(node, xs)) -> app(c_app(flatwithsub, f), xs) [1] app(c_app(flatwith, f), c_app(node, xs)) -> app(null_app, xs) [1] app(c_app(flatwith, c_app(append, c_app(c_app(cons, x'), xs'))), c_app(leaf, x)) -> app(app(cons, app(app(cons, x'), app(app(append, xs'), x))), nil) [2] app(c_app(flatwith, c_app(flatwithsub, f')), c_app(leaf, nil)) -> app(app(cons, nil), nil) [2] app(c_app(flatwith, c_app(append, nil)), c_app(leaf, x)) -> app(app(cons, x), nil) [2] app(c_app(flatwith, c_app(flatwithsub, f'')), c_app(leaf, c_app(c_app(cons, x''), xs''))) -> app(app(cons, app(app(append, app(app(flatwith, f''), x'')), app(app(flatwithsub, f''), xs''))), nil) [2] app(c_app(flatwith, c_app(flatwith, f1)), c_app(leaf, c_app(node, xs1))) -> app(app(cons, app(app(flatwithsub, f1), xs1)), nil) [2] app(c_app(flatwith, c_app(flatwith, f2)), c_app(leaf, c_app(leaf, x2))) -> app(app(cons, app(app(cons, app(f2, x2)), nil)), nil) [2] app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, c_app(f, x)), nil) [1] app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, null_app), nil) [1] encArg(append) -> append [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(flatwith) -> flatwith [0] encArg(leaf) -> leaf [0] encArg(node) -> node [0] encArg(flatwithsub) -> flatwithsub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_append -> append [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_flatwith -> flatwith [0] encode_leaf -> leaf [0] encode_node -> node [0] encode_flatwithsub -> flatwithsub [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_append -> null_encode_append [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_flatwith -> null_encode_flatwith [0] encode_leaf -> null_encode_leaf [0] encode_node -> null_encode_node [0] encode_flatwithsub -> null_encode_flatwithsub [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app c_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app cons_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub: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: append => 0 cons => 1 flatwithsub => 3 nil => 5 flatwith => 2 node => 6 leaf => 4 null_encArg => 0 null_encode_app => 0 null_encode_append => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_flatwith => 0 null_encode_leaf => 0 null_encode_node => 0 null_encode_flatwithsub => 0 null_app => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> ys :|: z = 1 + 0 + 5, z' = ys, ys >= 0 app(z, z') -{ 2 }-> app(app(1, x), 5) :|: x >= 0, z = 1 + 2 + (1 + 0 + 5), z' = 1 + 4 + x app(z, z') -{ 2 }-> app(app(1, app(app(3, f1), xs1)), 5) :|: z = 1 + 2 + (1 + 2 + f1), xs1 >= 0, f1 >= 0, z' = 1 + 4 + (1 + 6 + xs1) app(z, z') -{ 2 }-> app(app(1, app(app(1, x'), app(app(0, xs'), x))), 5) :|: x' >= 0, xs' >= 0, x >= 0, z = 1 + 2 + (1 + 0 + (1 + (1 + 1 + x') + xs')), z' = 1 + 4 + x app(z, z') -{ 2 }-> app(app(1, app(app(1, app(f2, x2)), 5)), 5) :|: z' = 1 + 4 + (1 + 4 + x2), f2 >= 0, z = 1 + 2 + (1 + 2 + f2), x2 >= 0 app(z, z') -{ 2 }-> app(app(1, app(app(0, app(app(2, f''), x'')), app(app(3, f''), xs''))), 5) :|: f'' >= 0, xs'' >= 0, z' = 1 + 4 + (1 + (1 + 1 + x'') + xs''), z = 1 + 2 + (1 + 3 + f''), x'' >= 0 app(z, z') -{ 2 }-> app(app(1, 5), 5) :|: z = 1 + 2 + (1 + 3 + f'), z' = 1 + 4 + 5, f' >= 0 app(z, z') -{ 1 }-> app(app(1, 0), 5) :|: f >= 0, z = 1 + 2 + f, x >= 0, z' = 1 + 4 + x app(z, z') -{ 1 }-> app(app(1, 1 + f + x), 5) :|: f >= 0, z = 1 + 2 + f, x >= 0, z' = 1 + 4 + x app(z, z') -{ 1 }-> app(app(0, app(0, x)), app(0, xs)) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(0, app(0, x)), app(1 + 3 + f, xs)) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(0, app(1 + 2 + f, x)), app(0, xs)) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(0, app(1 + 2 + f, x)), app(1 + 3 + f, xs)) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(0, xs) :|: f >= 0, xs >= 0, z = 1 + 2 + f, z' = 1 + 6 + xs app(z, z') -{ 1 }-> app(0, app(0, ys)) :|: xs >= 0, z' = ys, ys >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(0, app(1 + 0 + xs, ys)) :|: xs >= 0, z' = ys, ys >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(1 + 3 + f, xs) :|: f >= 0, xs >= 0, z = 1 + 2 + f, z' = 1 + 6 + xs app(z, z') -{ 1 }-> app(1 + 1 + x, app(0, ys)) :|: xs >= 0, z' = ys, ys >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(1 + 1 + x, app(1 + 0 + xs, ys)) :|: xs >= 0, z' = ys, ys >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> 5 :|: z = 1 + 3 + f, f >= 0, z' = 5 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_append -{ 0 }-> 0 :|: encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_flatwith -{ 0 }-> 2 :|: encode_flatwith -{ 0 }-> 0 :|: encode_flatwithsub -{ 0 }-> 3 :|: encode_flatwithsub -{ 0 }-> 0 :|: encode_leaf -{ 0 }-> 4 :|: encode_leaf -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 5 :|: encode_nil -{ 0 }-> 0 :|: encode_node -{ 0 }-> 6 :|: encode_node -{ 0 }-> 0 :|: ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> z' :|: z = 1 + 0 + 5, z' >= 0 app(z, z') -{ 2 }-> app(app(1, app(app(3, z - 6), z' - 12)), 5) :|: z' - 12 >= 0, z - 6 >= 0 app(z, z') -{ 2 }-> app(app(1, app(app(1, x'), app(app(0, xs'), z' - 5))), 5) :|: x' >= 0, xs' >= 0, z' - 5 >= 0, z = 1 + 2 + (1 + 0 + (1 + (1 + 1 + x') + xs')) app(z, z') -{ 2 }-> app(app(1, app(app(1, app(z - 6, z' - 10)), 5)), 5) :|: z - 6 >= 0, z' - 10 >= 0 app(z, z') -{ 2 }-> app(app(1, app(app(0, app(app(2, z - 7), x'')), app(app(3, z - 7), xs''))), 5) :|: z - 7 >= 0, xs'' >= 0, z' = 1 + 4 + (1 + (1 + 1 + x'') + xs''), x'' >= 0 app(z, z') -{ 2 }-> app(app(1, 5), 5) :|: z' = 1 + 4 + 5, z - 7 >= 0 app(z, z') -{ 1 }-> app(app(1, 0), 5) :|: z - 3 >= 0, z' - 5 >= 0 app(z, z') -{ 2 }-> app(app(1, z' - 5), 5) :|: z' - 5 >= 0, z = 1 + 2 + (1 + 0 + 5) app(z, z') -{ 1 }-> app(app(1, 1 + (z - 3) + (z' - 5)), 5) :|: z - 3 >= 0, z' - 5 >= 0 app(z, z') -{ 1 }-> app(app(0, app(0, x)), app(0, xs)) :|: z - 4 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(0, app(0, x)), app(1 + 3 + (z - 4), xs)) :|: z - 4 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(0, app(1 + 2 + (z - 4), x)), app(0, xs)) :|: z - 4 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(app(0, app(1 + 2 + (z - 4), x)), app(1 + 3 + (z - 4), xs)) :|: z - 4 >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> app(0, app(0, z')) :|: xs >= 0, z' >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(0, app(1 + 0 + xs, z')) :|: xs >= 0, z' >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(0, z' - 7) :|: z - 3 >= 0, z' - 7 >= 0 app(z, z') -{ 1 }-> app(1 + 3 + (z - 3), z' - 7) :|: z - 3 >= 0, z' - 7 >= 0 app(z, z') -{ 1 }-> app(1 + 1 + x, app(0, z')) :|: xs >= 0, z' >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(1 + 1 + x, app(1 + 0 + xs, z')) :|: xs >= 0, z' >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> 5 :|: z - 4 >= 0, z' = 5 app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 app(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_app(z, z') -{ 0 }-> app(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_append -{ 0 }-> 0 :|: encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_flatwith -{ 0 }-> 2 :|: encode_flatwith -{ 0 }-> 0 :|: encode_flatwithsub -{ 0 }-> 3 :|: encode_flatwithsub -{ 0 }-> 0 :|: encode_leaf -{ 0 }-> 4 :|: encode_leaf -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 5 :|: encode_nil -{ 0 }-> 0 :|: encode_node -{ 0 }-> 6 :|: encode_node -{ 0 }-> 0 :|: ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_append -> null_encode_append [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_flatwith -> null_encode_flatwith [0] encode_leaf -> null_encode_leaf [0] encode_node -> null_encode_node [0] encode_flatwithsub -> null_encode_flatwithsub [0] app(v0, v1) -> null_app [0] And the following fresh constants: null_encArg, null_encode_app, null_encode_append, null_encode_nil, null_encode_cons, null_encode_flatwith, null_encode_leaf, null_encode_node, null_encode_flatwithsub, null_app ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: app(c_app(append, c_app(c_app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys)) [1] app(c_app(flatwithsub, f), nil) -> nil [1] app(c_app(append, nil), ys) -> ys [1] app(c_app(flatwithsub, f), c_app(c_app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs)) [1] app(c_app(flatwith, f), c_app(node, xs)) -> app(app(flatwithsub, f), xs) [1] app(c_app(flatwith, f), c_app(leaf, x)) -> app(app(cons, app(f, x)), nil) [1] encArg(append) -> append [0] encArg(nil) -> nil [0] encArg(cons) -> cons [0] encArg(flatwith) -> flatwith [0] encArg(leaf) -> leaf [0] encArg(node) -> node [0] encArg(flatwithsub) -> flatwithsub [0] encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) [0] encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) [0] encode_append -> append [0] encode_nil -> nil [0] encode_cons -> cons [0] encode_flatwith -> flatwith [0] encode_leaf -> leaf [0] encode_node -> node [0] encode_flatwithsub -> flatwithsub [0] app(x0, x1) -> c_app(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_app(v0, v1) -> null_encode_app [0] encode_append -> null_encode_append [0] encode_nil -> null_encode_nil [0] encode_cons -> null_encode_cons [0] encode_flatwith -> null_encode_flatwith [0] encode_leaf -> null_encode_leaf [0] encode_node -> null_encode_node [0] encode_flatwithsub -> null_encode_flatwithsub [0] app(v0, v1) -> null_app [0] The TRS has the following type information: app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app c_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app cons_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app -> append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encArg :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_append :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_nil :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_cons :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_flatwith :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_leaf :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_node :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_encode_flatwithsub :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub:null_app null_app :: append:cons:c_app:flatwithsub:nil:flatwith:node:leaf:cons_app:null_encArg:null_encode_app:null_encode_append:null_encode_nil:null_encode_cons:null_encode_flatwith:null_encode_leaf:null_encode_node:null_encode_flatwithsub: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: append => 0 cons => 1 flatwithsub => 3 nil => 5 flatwith => 2 node => 6 leaf => 4 null_encArg => 0 null_encode_app => 0 null_encode_append => 0 null_encode_nil => 0 null_encode_cons => 0 null_encode_flatwith => 0 null_encode_leaf => 0 null_encode_node => 0 null_encode_flatwithsub => 0 null_app => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: app(z, z') -{ 1 }-> ys :|: z = 1 + 0 + 5, z' = ys, ys >= 0 app(z, z') -{ 1 }-> app(app(3, f), xs) :|: f >= 0, xs >= 0, z = 1 + 2 + f, z' = 1 + 6 + xs app(z, z') -{ 1 }-> app(app(1, x), app(app(0, xs), ys)) :|: xs >= 0, z' = ys, ys >= 0, x >= 0, z = 1 + 0 + (1 + (1 + 1 + x) + xs) app(z, z') -{ 1 }-> app(app(1, app(f, x)), 5) :|: f >= 0, z = 1 + 2 + f, x >= 0, z' = 1 + 4 + x app(z, z') -{ 1 }-> app(app(0, app(app(2, f), x)), app(app(3, f), xs)) :|: z = 1 + 3 + f, f >= 0, xs >= 0, x >= 0, z' = 1 + (1 + 1 + x) + xs app(z, z') -{ 1 }-> 5 :|: z = 1 + 3 + f, f >= 0, z' = 5 app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 app(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 6 :|: z = 6 encArg(z) -{ 0 }-> 5 :|: z = 5 encArg(z) -{ 0 }-> 4 :|: z = 4 encArg(z) -{ 0 }-> 3 :|: z = 3 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_app(z, z') -{ 0 }-> app(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_append -{ 0 }-> 0 :|: encode_cons -{ 0 }-> 1 :|: encode_cons -{ 0 }-> 0 :|: encode_flatwith -{ 0 }-> 2 :|: encode_flatwith -{ 0 }-> 0 :|: encode_flatwithsub -{ 0 }-> 3 :|: encode_flatwithsub -{ 0 }-> 0 :|: encode_leaf -{ 0 }-> 4 :|: encode_leaf -{ 0 }-> 0 :|: encode_nil -{ 0 }-> 5 :|: encode_nil -{ 0 }-> 0 :|: encode_node -{ 0 }-> 6 :|: encode_node -{ 0 }-> 0 :|: Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: ENCARG(append) -> c ENCARG(nil) -> c1 ENCARG(cons) -> c2 ENCARG(flatwith) -> c3 ENCARG(leaf) -> c4 ENCARG(node) -> c5 ENCARG(flatwithsub) -> c6 ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c8(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APPEND -> c9 ENCODE_NIL -> c10 ENCODE_CONS -> c11 ENCODE_FLATWITH -> c12 ENCODE_LEAF -> c13 ENCODE_NODE -> c14 ENCODE_FLATWITHSUB -> c15 APP(app(append, nil), z0) -> c16 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(cons, z0), app(app(append, z1), z2)), APP(cons, z0), APP(app(append, z1), z2), APP(append, z1)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(app(cons, app(z0, z1)), nil), APP(cons, app(z0, z1)), APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1), APP(flatwithsub, z0)) APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(append, app(app(flatwith, z0), z1)), APP(app(flatwith, z0), z1), APP(flatwith, z0), APP(app(flatwithsub, z0), z2), APP(flatwithsub, z0)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(cons, z0), app(app(append, z1), z2)), APP(cons, z0), APP(app(append, z1), z2), APP(append, z1)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(app(cons, app(z0, z1)), nil), APP(cons, app(z0, z1)), APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1), APP(flatwithsub, z0)) APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(append, app(app(flatwith, z0), z1)), APP(app(flatwith, z0), z1), APP(flatwith, z0), APP(app(flatwithsub, z0), z2), APP(flatwithsub, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_append, encode_nil, encode_cons, encode_flatwith, encode_leaf, encode_node, encode_flatwithsub Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, ENCODE_APPEND, ENCODE_NIL, ENCODE_CONS, ENCODE_FLATWITH, ENCODE_LEAF, ENCODE_NODE, ENCODE_FLATWITHSUB, APP_2 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7_3, c8_3, c9, c10, c11, c12, c13, c14, c15, c16, c17_4, c18_3, c19_2, c20, c21_6 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 14 trailing nodes: ENCARG(node) -> c5 ENCODE_FLATWITH -> c12 ENCODE_NODE -> c14 ENCODE_FLATWITHSUB -> c15 ENCARG(append) -> c ENCODE_LEAF -> c13 ENCODE_NIL -> c10 ENCARG(nil) -> c1 ENCARG(flatwithsub) -> c6 ENCARG(cons) -> c2 ENCODE_APPEND -> c9 ENCARG(leaf) -> c4 ENCODE_CONS -> c11 ENCARG(flatwith) -> c3 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c8(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(append, nil), z0) -> c16 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(cons, z0), app(app(append, z1), z2)), APP(cons, z0), APP(app(append, z1), z2), APP(append, z1)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(app(cons, app(z0, z1)), nil), APP(cons, app(z0, z1)), APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1), APP(flatwithsub, z0)) APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(append, app(app(flatwith, z0), z1)), APP(app(flatwith, z0), z1), APP(flatwith, z0), APP(app(flatwithsub, z0), z2), APP(flatwithsub, z0)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(cons, z0), app(app(append, z1), z2)), APP(cons, z0), APP(app(append, z1), z2), APP(append, z1)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(app(cons, app(z0, z1)), nil), APP(cons, app(z0, z1)), APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1), APP(flatwithsub, z0)) APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(append, app(app(flatwith, z0), z1)), APP(app(flatwith, z0), z1), APP(flatwith, z0), APP(app(flatwithsub, z0), z2), APP(flatwithsub, z0)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_append, encode_nil, encode_cons, encode_flatwith, encode_leaf, encode_node, encode_flatwithsub Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c7_3, c8_3, c16, c17_4, c18_3, c19_2, c20, c21_6 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_APP(z0, z1) -> c8(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_append, encode_nil, encode_cons, encode_flatwith, encode_leaf, encode_node, encode_flatwithsub Defined Pair Symbols: ENCARG_1, ENCODE_APP_2, APP_2 Compound Symbols: c7_3, c8_3, c16, c20, c17_1, c18_1, c19_1, c21_3 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCODE_APP(z0, z1) -> c(ENCARG(z0)) ENCODE_APP(z0, z1) -> c(ENCARG(z1)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_append, encode_nil, encode_cons, encode_flatwith, encode_leaf, encode_node, encode_flatwithsub Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c7_3, c16, c20, c17_1, c18_1, c19_1, c21_3, 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(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) encode_app(z0, z1) -> app(encArg(z0), encArg(z1)) encode_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) K tuples:none Defined Rule Symbols: app_2, encArg_1, encode_app_2, encode_append, encode_nil, encode_cons, encode_flatwith, encode_leaf, encode_node, encode_flatwithsub Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c7_3, c16, c20, c17_1, c18_1, c19_1, c21_3, 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_append -> append encode_nil -> nil encode_cons -> cons encode_flatwith -> flatwith encode_leaf -> leaf encode_node -> node encode_flatwithsub -> flatwithsub ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: ENCARG_1, APP_2, ENCODE_APP_2 Compound Symbols: c7_3, c16, c20, c17_1, c18_1, c19_1, c21_3, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(z0, z1)) -> c7(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0), ENCARG(append)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0), ENCARG(flatwith)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0), ENCARG(leaf)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0), ENCARG(node)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0), ENCARG(flatwithsub)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(append, x1)) -> c7(APP(append, encArg(x1)), ENCARG(append), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(APP(flatwith, encArg(x1)), ENCARG(flatwith), ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(APP(leaf, encArg(x1)), ENCARG(leaf), ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(APP(node, encArg(x1)), ENCARG(node), ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(APP(flatwithsub, encArg(x1)), ENCARG(flatwithsub), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0), ENCARG(append)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0), ENCARG(nil)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0), ENCARG(cons)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0), ENCARG(flatwith)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0), ENCARG(leaf)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0), ENCARG(node)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0), ENCARG(flatwithsub)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(append, x1)) -> c7(APP(append, encArg(x1)), ENCARG(append), ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(APP(nil, encArg(x1)), ENCARG(nil), ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(APP(cons, encArg(x1)), ENCARG(cons), ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(APP(flatwith, encArg(x1)), ENCARG(flatwith), ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(APP(leaf, encArg(x1)), ENCARG(leaf), ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(APP(node, encArg(x1)), ENCARG(node), ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(APP(flatwithsub, encArg(x1)), ENCARG(flatwithsub), ENCARG(x1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c21_3, c_1, c7_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 21 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c21_3, c_1, c7_3, c7_2, c7_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(flatwithsub, z0), app(app(cons, z1), z2)) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), z2)) by APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCODE_APP_2, ENCARG_1 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c_1, c7_3, c7_2, c7_1, c21_3 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1))) by ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(append, x1) -> c(APP(append, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(flatwith, x1) -> c(APP(flatwith, encArg(x1))) ENCODE_APP(leaf, x1) -> c(APP(leaf, encArg(x1))) ENCODE_APP(node, x1) -> c(APP(node, encArg(x1))) ENCODE_APP(flatwithsub, x1) -> c(APP(flatwithsub, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(append, x1) -> c(APP(append, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(flatwith, x1) -> c(APP(flatwith, encArg(x1))) ENCODE_APP(leaf, x1) -> c(APP(leaf, encArg(x1))) ENCODE_APP(node, x1) -> c(APP(node, encArg(x1))) ENCODE_APP(flatwithsub, x1) -> c(APP(flatwithsub, encArg(x1))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_3, c7_2, c7_1, c21_3, c_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(flatwith, x1) -> c(APP(flatwith, encArg(x1))) ENCODE_APP(cons, x1) -> c(APP(cons, encArg(x1))) ENCODE_APP(leaf, x1) -> c(APP(leaf, encArg(x1))) ENCODE_APP(append, x1) -> c(APP(append, encArg(x1))) ENCODE_APP(node, x1) -> c(APP(node, encArg(x1))) ENCODE_APP(nil, x1) -> c(APP(nil, encArg(x1))) ENCODE_APP(flatwithsub, x1) -> c(APP(flatwithsub, encArg(x1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_3, c7_2, c7_1, c21_3, c_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons_app(z0, z1))) -> c7(APP(encArg(x0), app(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_app(z0, z1))) by ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(APP(append, app(encArg(x1), encArg(x2))), ENCARG(append), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(APP(flatwith, app(encArg(x1), encArg(x2))), ENCARG(flatwith), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(APP(leaf, app(encArg(x1), encArg(x2))), ENCARG(leaf), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(APP(node, app(encArg(x1), encArg(x2))), ENCARG(node), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(APP(flatwithsub, app(encArg(x1), encArg(x2))), ENCARG(flatwithsub), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(APP(append, app(encArg(x1), encArg(x2))), ENCARG(append), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(APP(nil, app(encArg(x1), encArg(x2))), ENCARG(nil), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(APP(cons, app(encArg(x1), encArg(x2))), ENCARG(cons), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(APP(flatwith, app(encArg(x1), encArg(x2))), ENCARG(flatwith), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(APP(leaf, app(encArg(x1), encArg(x2))), ENCARG(leaf), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(APP(node, app(encArg(x1), encArg(x2))), ENCARG(node), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(APP(flatwithsub, app(encArg(x1), encArg(x2))), ENCARG(flatwithsub), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_3, c7_2, c7_1, c21_3, c_1 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 14 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_3, c7_2, c7_1, c21_3, c_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(cons_app(z0, z1), x1)) -> c7(APP(app(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_app(z0, z1)), ENCARG(x1)) by ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1)), ENCARG(append)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1)), ENCARG(flatwith)) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1)), ENCARG(leaf)) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1)), ENCARG(node)) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1)), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(x0, x1), cons_app(z0, z1))) -> c7(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, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(APP(app(leaf, encArg(x1)), encArg(x2)), ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(APP(app(node, encArg(x1)), encArg(x2)), ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1)), ENCARG(append)) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1)), ENCARG(nil)) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1)), ENCARG(cons)) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1)), ENCARG(flatwith)) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1)), ENCARG(leaf)) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1)), ENCARG(node)) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1)), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(APP(app(nil, encArg(x1)), encArg(x2)), ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(APP(app(cons, encArg(x1)), encArg(x2)), ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(APP(app(leaf, encArg(x1)), encArg(x2)), ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(APP(app(node, encArg(x1)), encArg(x2)), ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, append)) -> c7(APP(encArg(x0), append), ENCARG(x0)) by ENCARG(cons_app(append, append)) -> c7(APP(append, append), ENCARG(append)) ENCARG(cons_app(nil, append)) -> c7(APP(nil, append), ENCARG(nil)) ENCARG(cons_app(cons, append)) -> c7(APP(cons, append), ENCARG(cons)) ENCARG(cons_app(flatwith, append)) -> c7(APP(flatwith, append), ENCARG(flatwith)) ENCARG(cons_app(leaf, append)) -> c7(APP(leaf, append), ENCARG(leaf)) ENCARG(cons_app(node, append)) -> c7(APP(node, append), ENCARG(node)) ENCARG(cons_app(flatwithsub, append)) -> c7(APP(flatwithsub, append), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), append)) -> c7(APP(app(encArg(z0), encArg(z1)), append), ENCARG(cons_app(z0, z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, append)) -> c7(APP(append, append), ENCARG(append)) ENCARG(cons_app(nil, append)) -> c7(APP(nil, append), ENCARG(nil)) ENCARG(cons_app(cons, append)) -> c7(APP(cons, append), ENCARG(cons)) ENCARG(cons_app(flatwith, append)) -> c7(APP(flatwith, append), ENCARG(flatwith)) ENCARG(cons_app(leaf, append)) -> c7(APP(leaf, append), ENCARG(leaf)) ENCARG(cons_app(node, append)) -> c7(APP(node, append), ENCARG(node)) ENCARG(cons_app(flatwithsub, append)) -> c7(APP(flatwithsub, append), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(leaf, append)) -> c7(APP(leaf, append), ENCARG(leaf)) ENCARG(cons_app(node, append)) -> c7(APP(node, append), ENCARG(node)) ENCARG(cons_app(cons, append)) -> c7(APP(cons, append), ENCARG(cons)) ENCARG(cons_app(flatwith, append)) -> c7(APP(flatwith, append), ENCARG(flatwith)) ENCARG(cons_app(flatwithsub, append)) -> c7(APP(flatwithsub, append), ENCARG(flatwithsub)) ENCARG(cons_app(nil, append)) -> c7(APP(nil, append), ENCARG(nil)) ENCARG(cons_app(append, append)) -> c7(APP(append, append), ENCARG(append)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, nil)) -> c7(APP(encArg(x0), nil), ENCARG(x0)) by ENCARG(cons_app(append, nil)) -> c7(APP(append, nil), ENCARG(append)) ENCARG(cons_app(nil, nil)) -> c7(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c7(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(flatwith, nil)) -> c7(APP(flatwith, nil), ENCARG(flatwith)) ENCARG(cons_app(leaf, nil)) -> c7(APP(leaf, nil), ENCARG(leaf)) ENCARG(cons_app(node, nil)) -> c7(APP(node, nil), ENCARG(node)) ENCARG(cons_app(flatwithsub, nil)) -> c7(APP(flatwithsub, nil), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), nil)) -> c7(APP(app(encArg(z0), encArg(z1)), nil), ENCARG(cons_app(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, nil)) -> c7(APP(append, nil), ENCARG(append)) ENCARG(cons_app(nil, nil)) -> c7(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(cons, nil)) -> c7(APP(cons, nil), ENCARG(cons)) ENCARG(cons_app(flatwith, nil)) -> c7(APP(flatwith, nil), ENCARG(flatwith)) ENCARG(cons_app(leaf, nil)) -> c7(APP(leaf, nil), ENCARG(leaf)) ENCARG(cons_app(node, nil)) -> c7(APP(node, nil), ENCARG(node)) ENCARG(cons_app(flatwithsub, nil)) -> c7(APP(flatwithsub, nil), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(flatwithsub, nil)) -> c7(APP(flatwithsub, nil), ENCARG(flatwithsub)) ENCARG(cons_app(node, nil)) -> c7(APP(node, nil), ENCARG(node)) ENCARG(cons_app(nil, nil)) -> c7(APP(nil, nil), ENCARG(nil)) ENCARG(cons_app(flatwith, nil)) -> c7(APP(flatwith, nil), ENCARG(flatwith)) ENCARG(cons_app(append, nil)) -> c7(APP(append, nil), ENCARG(append)) ENCARG(cons_app(leaf, nil)) -> c7(APP(leaf, nil), ENCARG(leaf)) ENCARG(cons_app(cons, nil)) -> c7(APP(cons, nil), ENCARG(cons)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, cons)) -> c7(APP(encArg(x0), cons), ENCARG(x0)) by ENCARG(cons_app(append, cons)) -> c7(APP(append, cons), ENCARG(append)) ENCARG(cons_app(nil, cons)) -> c7(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c7(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(flatwith, cons)) -> c7(APP(flatwith, cons), ENCARG(flatwith)) ENCARG(cons_app(leaf, cons)) -> c7(APP(leaf, cons), ENCARG(leaf)) ENCARG(cons_app(node, cons)) -> c7(APP(node, cons), ENCARG(node)) ENCARG(cons_app(flatwithsub, cons)) -> c7(APP(flatwithsub, cons), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), cons)) -> c7(APP(app(encArg(z0), encArg(z1)), cons), ENCARG(cons_app(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, cons)) -> c7(APP(append, cons), ENCARG(append)) ENCARG(cons_app(nil, cons)) -> c7(APP(nil, cons), ENCARG(nil)) ENCARG(cons_app(cons, cons)) -> c7(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(flatwith, cons)) -> c7(APP(flatwith, cons), ENCARG(flatwith)) ENCARG(cons_app(leaf, cons)) -> c7(APP(leaf, cons), ENCARG(leaf)) ENCARG(cons_app(node, cons)) -> c7(APP(node, cons), ENCARG(node)) ENCARG(cons_app(flatwithsub, cons)) -> c7(APP(flatwithsub, cons), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(flatwithsub, cons)) -> c7(APP(flatwithsub, cons), ENCARG(flatwithsub)) ENCARG(cons_app(node, cons)) -> c7(APP(node, cons), ENCARG(node)) ENCARG(cons_app(cons, cons)) -> c7(APP(cons, cons), ENCARG(cons)) ENCARG(cons_app(leaf, cons)) -> c7(APP(leaf, cons), ENCARG(leaf)) ENCARG(cons_app(flatwith, cons)) -> c7(APP(flatwith, cons), ENCARG(flatwith)) ENCARG(cons_app(append, cons)) -> c7(APP(append, cons), ENCARG(append)) ENCARG(cons_app(nil, cons)) -> c7(APP(nil, cons), ENCARG(nil)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, flatwith)) -> c7(APP(encArg(x0), flatwith), ENCARG(x0)) by ENCARG(cons_app(append, flatwith)) -> c7(APP(append, flatwith), ENCARG(append)) ENCARG(cons_app(nil, flatwith)) -> c7(APP(nil, flatwith), ENCARG(nil)) ENCARG(cons_app(cons, flatwith)) -> c7(APP(cons, flatwith), ENCARG(cons)) ENCARG(cons_app(flatwith, flatwith)) -> c7(APP(flatwith, flatwith), ENCARG(flatwith)) ENCARG(cons_app(leaf, flatwith)) -> c7(APP(leaf, flatwith), ENCARG(leaf)) ENCARG(cons_app(node, flatwith)) -> c7(APP(node, flatwith), ENCARG(node)) ENCARG(cons_app(flatwithsub, flatwith)) -> c7(APP(flatwithsub, flatwith), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), flatwith)) -> c7(APP(app(encArg(z0), encArg(z1)), flatwith), ENCARG(cons_app(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, flatwith)) -> c7(APP(append, flatwith), ENCARG(append)) ENCARG(cons_app(nil, flatwith)) -> c7(APP(nil, flatwith), ENCARG(nil)) ENCARG(cons_app(cons, flatwith)) -> c7(APP(cons, flatwith), ENCARG(cons)) ENCARG(cons_app(flatwith, flatwith)) -> c7(APP(flatwith, flatwith), ENCARG(flatwith)) ENCARG(cons_app(leaf, flatwith)) -> c7(APP(leaf, flatwith), ENCARG(leaf)) ENCARG(cons_app(node, flatwith)) -> c7(APP(node, flatwith), ENCARG(node)) ENCARG(cons_app(flatwithsub, flatwith)) -> c7(APP(flatwithsub, flatwith), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(flatwith, flatwith)) -> c7(APP(flatwith, flatwith), ENCARG(flatwith)) ENCARG(cons_app(cons, flatwith)) -> c7(APP(cons, flatwith), ENCARG(cons)) ENCARG(cons_app(nil, flatwith)) -> c7(APP(nil, flatwith), ENCARG(nil)) ENCARG(cons_app(node, flatwith)) -> c7(APP(node, flatwith), ENCARG(node)) ENCARG(cons_app(leaf, flatwith)) -> c7(APP(leaf, flatwith), ENCARG(leaf)) ENCARG(cons_app(append, flatwith)) -> c7(APP(append, flatwith), ENCARG(append)) ENCARG(cons_app(flatwithsub, flatwith)) -> c7(APP(flatwithsub, flatwith), ENCARG(flatwithsub)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, leaf)) -> c7(APP(encArg(x0), leaf), ENCARG(x0)) by ENCARG(cons_app(append, leaf)) -> c7(APP(append, leaf), ENCARG(append)) ENCARG(cons_app(nil, leaf)) -> c7(APP(nil, leaf), ENCARG(nil)) ENCARG(cons_app(cons, leaf)) -> c7(APP(cons, leaf), ENCARG(cons)) ENCARG(cons_app(flatwith, leaf)) -> c7(APP(flatwith, leaf), ENCARG(flatwith)) ENCARG(cons_app(leaf, leaf)) -> c7(APP(leaf, leaf), ENCARG(leaf)) ENCARG(cons_app(node, leaf)) -> c7(APP(node, leaf), ENCARG(node)) ENCARG(cons_app(flatwithsub, leaf)) -> c7(APP(flatwithsub, leaf), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), leaf)) -> c7(APP(app(encArg(z0), encArg(z1)), leaf), ENCARG(cons_app(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, leaf)) -> c7(APP(append, leaf), ENCARG(append)) ENCARG(cons_app(nil, leaf)) -> c7(APP(nil, leaf), ENCARG(nil)) ENCARG(cons_app(cons, leaf)) -> c7(APP(cons, leaf), ENCARG(cons)) ENCARG(cons_app(flatwith, leaf)) -> c7(APP(flatwith, leaf), ENCARG(flatwith)) ENCARG(cons_app(leaf, leaf)) -> c7(APP(leaf, leaf), ENCARG(leaf)) ENCARG(cons_app(node, leaf)) -> c7(APP(node, leaf), ENCARG(node)) ENCARG(cons_app(flatwithsub, leaf)) -> c7(APP(flatwithsub, leaf), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(flatwithsub, leaf)) -> c7(APP(flatwithsub, leaf), ENCARG(flatwithsub)) ENCARG(cons_app(nil, leaf)) -> c7(APP(nil, leaf), ENCARG(nil)) ENCARG(cons_app(flatwith, leaf)) -> c7(APP(flatwith, leaf), ENCARG(flatwith)) ENCARG(cons_app(node, leaf)) -> c7(APP(node, leaf), ENCARG(node)) ENCARG(cons_app(cons, leaf)) -> c7(APP(cons, leaf), ENCARG(cons)) ENCARG(cons_app(leaf, leaf)) -> c7(APP(leaf, leaf), ENCARG(leaf)) ENCARG(cons_app(append, leaf)) -> c7(APP(append, leaf), ENCARG(append)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, node)) -> c7(APP(encArg(x0), node), ENCARG(x0)) by ENCARG(cons_app(append, node)) -> c7(APP(append, node), ENCARG(append)) ENCARG(cons_app(nil, node)) -> c7(APP(nil, node), ENCARG(nil)) ENCARG(cons_app(cons, node)) -> c7(APP(cons, node), ENCARG(cons)) ENCARG(cons_app(flatwith, node)) -> c7(APP(flatwith, node), ENCARG(flatwith)) ENCARG(cons_app(leaf, node)) -> c7(APP(leaf, node), ENCARG(leaf)) ENCARG(cons_app(node, node)) -> c7(APP(node, node), ENCARG(node)) ENCARG(cons_app(flatwithsub, node)) -> c7(APP(flatwithsub, node), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), node)) -> c7(APP(app(encArg(z0), encArg(z1)), node), ENCARG(cons_app(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, node)) -> c7(APP(append, node), ENCARG(append)) ENCARG(cons_app(nil, node)) -> c7(APP(nil, node), ENCARG(nil)) ENCARG(cons_app(cons, node)) -> c7(APP(cons, node), ENCARG(cons)) ENCARG(cons_app(flatwith, node)) -> c7(APP(flatwith, node), ENCARG(flatwith)) ENCARG(cons_app(leaf, node)) -> c7(APP(leaf, node), ENCARG(leaf)) ENCARG(cons_app(node, node)) -> c7(APP(node, node), ENCARG(node)) ENCARG(cons_app(flatwithsub, node)) -> c7(APP(flatwithsub, node), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(cons, node)) -> c7(APP(cons, node), ENCARG(cons)) ENCARG(cons_app(nil, node)) -> c7(APP(nil, node), ENCARG(nil)) ENCARG(cons_app(flatwithsub, node)) -> c7(APP(flatwithsub, node), ENCARG(flatwithsub)) ENCARG(cons_app(leaf, node)) -> c7(APP(leaf, node), ENCARG(leaf)) ENCARG(cons_app(append, node)) -> c7(APP(append, node), ENCARG(append)) ENCARG(cons_app(flatwith, node)) -> c7(APP(flatwith, node), ENCARG(flatwith)) ENCARG(cons_app(node, node)) -> c7(APP(node, node), ENCARG(node)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_2, c7_1, c21_3, c_1, c7_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_app(x0, flatwithsub)) -> c7(APP(encArg(x0), flatwithsub), ENCARG(x0)) by ENCARG(cons_app(append, flatwithsub)) -> c7(APP(append, flatwithsub), ENCARG(append)) ENCARG(cons_app(nil, flatwithsub)) -> c7(APP(nil, flatwithsub), ENCARG(nil)) ENCARG(cons_app(cons, flatwithsub)) -> c7(APP(cons, flatwithsub), ENCARG(cons)) ENCARG(cons_app(flatwith, flatwithsub)) -> c7(APP(flatwith, flatwithsub), ENCARG(flatwith)) ENCARG(cons_app(leaf, flatwithsub)) -> c7(APP(leaf, flatwithsub), ENCARG(leaf)) ENCARG(cons_app(node, flatwithsub)) -> c7(APP(node, flatwithsub), ENCARG(node)) ENCARG(cons_app(flatwithsub, flatwithsub)) -> c7(APP(flatwithsub, flatwithsub), ENCARG(flatwithsub)) ENCARG(cons_app(cons_app(z0, z1), flatwithsub)) -> c7(APP(app(encArg(z0), encArg(z1)), flatwithsub), ENCARG(cons_app(z0, z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) ENCARG(cons_app(append, flatwithsub)) -> c7(APP(append, flatwithsub), ENCARG(append)) ENCARG(cons_app(nil, flatwithsub)) -> c7(APP(nil, flatwithsub), ENCARG(nil)) ENCARG(cons_app(cons, flatwithsub)) -> c7(APP(cons, flatwithsub), ENCARG(cons)) ENCARG(cons_app(flatwith, flatwithsub)) -> c7(APP(flatwith, flatwithsub), ENCARG(flatwith)) ENCARG(cons_app(leaf, flatwithsub)) -> c7(APP(leaf, flatwithsub), ENCARG(leaf)) ENCARG(cons_app(node, flatwithsub)) -> c7(APP(node, flatwithsub), ENCARG(node)) ENCARG(cons_app(flatwithsub, flatwithsub)) -> c7(APP(flatwithsub, flatwithsub), ENCARG(flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCARG(cons_app(cons, flatwithsub)) -> c7(APP(cons, flatwithsub), ENCARG(cons)) ENCARG(cons_app(flatwithsub, flatwithsub)) -> c7(APP(flatwithsub, flatwithsub), ENCARG(flatwithsub)) ENCARG(cons_app(nil, flatwithsub)) -> c7(APP(nil, flatwithsub), ENCARG(nil)) ENCARG(cons_app(append, flatwithsub)) -> c7(APP(append, flatwithsub), ENCARG(append)) ENCARG(cons_app(node, flatwithsub)) -> c7(APP(node, flatwithsub), ENCARG(node)) ENCARG(cons_app(leaf, flatwithsub)) -> c7(APP(leaf, flatwithsub), ENCARG(leaf)) ENCARG(cons_app(flatwith, flatwithsub)) -> c7(APP(flatwith, flatwithsub), ENCARG(flatwith)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(flatwithsub, z0), app(app(cons, x1), nil)) -> c21(APP(app(append, app(app(flatwith, z0), x1)), nil), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), nil)) by APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) by APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x2)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), x2)) by APP(app(flatwithsub, z0), app(app(cons, app(node, x1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), x1)), nil), APP(app(flatwith, z0), app(node, x1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, x1)), app(app(cons, z1), z2))) -> c21(APP(app(append, app(app(flatwithsub, z0), x1)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), APP(app(flatwith, z0), app(node, x1)), APP(app(flatwithsub, z0), app(app(cons, z1), z2))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, append) -> c(APP(encArg(x0), append)) by ENCODE_APP(append, append) -> c(APP(append, append)) ENCODE_APP(nil, append) -> c(APP(nil, append)) ENCODE_APP(cons, append) -> c(APP(cons, append)) ENCODE_APP(flatwith, append) -> c(APP(flatwith, append)) ENCODE_APP(leaf, append) -> c(APP(leaf, append)) ENCODE_APP(node, append) -> c(APP(node, append)) ENCODE_APP(flatwithsub, append) -> c(APP(flatwithsub, append)) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(append, append) -> c(APP(append, append)) ENCODE_APP(nil, append) -> c(APP(nil, append)) ENCODE_APP(cons, append) -> c(APP(cons, append)) ENCODE_APP(flatwith, append) -> c(APP(flatwith, append)) ENCODE_APP(leaf, append) -> c(APP(leaf, append)) ENCODE_APP(node, append) -> c(APP(node, append)) ENCODE_APP(flatwithsub, append) -> c(APP(flatwithsub, append)) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(append, append) -> c(APP(append, append)) ENCODE_APP(leaf, append) -> c(APP(leaf, append)) ENCODE_APP(cons, append) -> c(APP(cons, append)) ENCODE_APP(flatwithsub, append) -> c(APP(flatwithsub, append)) ENCODE_APP(node, append) -> c(APP(node, append)) ENCODE_APP(flatwith, append) -> c(APP(flatwith, append)) ENCODE_APP(nil, append) -> c(APP(nil, append)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, nil) -> c(APP(encArg(x0), nil)) by ENCODE_APP(append, nil) -> c(APP(append, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(flatwith, nil) -> c(APP(flatwith, nil)) ENCODE_APP(leaf, nil) -> c(APP(leaf, nil)) ENCODE_APP(node, nil) -> c(APP(node, nil)) ENCODE_APP(flatwithsub, nil) -> c(APP(flatwithsub, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(append, nil) -> c(APP(append, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(flatwith, nil) -> c(APP(flatwith, nil)) ENCODE_APP(leaf, nil) -> c(APP(leaf, nil)) ENCODE_APP(node, nil) -> c(APP(node, nil)) ENCODE_APP(flatwithsub, nil) -> c(APP(flatwithsub, nil)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(append, nil) -> c(APP(append, nil)) ENCODE_APP(nil, nil) -> c(APP(nil, nil)) ENCODE_APP(flatwithsub, nil) -> c(APP(flatwithsub, nil)) ENCODE_APP(node, nil) -> c(APP(node, nil)) ENCODE_APP(cons, nil) -> c(APP(cons, nil)) ENCODE_APP(flatwith, nil) -> c(APP(flatwith, nil)) ENCODE_APP(leaf, nil) -> c(APP(leaf, nil)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, cons) -> c(APP(encArg(x0), cons)) by ENCODE_APP(append, cons) -> c(APP(append, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(flatwith, cons) -> c(APP(flatwith, cons)) ENCODE_APP(leaf, cons) -> c(APP(leaf, cons)) ENCODE_APP(node, cons) -> c(APP(node, cons)) ENCODE_APP(flatwithsub, cons) -> c(APP(flatwithsub, cons)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(append, cons) -> c(APP(append, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ENCODE_APP(flatwith, cons) -> c(APP(flatwith, cons)) ENCODE_APP(leaf, cons) -> c(APP(leaf, cons)) ENCODE_APP(node, cons) -> c(APP(node, cons)) ENCODE_APP(flatwithsub, cons) -> c(APP(flatwithsub, cons)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(flatwith, cons) -> c(APP(flatwith, cons)) ENCODE_APP(leaf, cons) -> c(APP(leaf, cons)) ENCODE_APP(append, cons) -> c(APP(append, cons)) ENCODE_APP(flatwithsub, cons) -> c(APP(flatwithsub, cons)) ENCODE_APP(nil, cons) -> c(APP(nil, cons)) ENCODE_APP(node, cons) -> c(APP(node, cons)) ENCODE_APP(cons, cons) -> c(APP(cons, cons)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, flatwith) -> c(APP(encArg(x0), flatwith)) by ENCODE_APP(append, flatwith) -> c(APP(append, flatwith)) ENCODE_APP(nil, flatwith) -> c(APP(nil, flatwith)) ENCODE_APP(cons, flatwith) -> c(APP(cons, flatwith)) ENCODE_APP(flatwith, flatwith) -> c(APP(flatwith, flatwith)) ENCODE_APP(leaf, flatwith) -> c(APP(leaf, flatwith)) ENCODE_APP(node, flatwith) -> c(APP(node, flatwith)) ENCODE_APP(flatwithsub, flatwith) -> c(APP(flatwithsub, flatwith)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(append, flatwith) -> c(APP(append, flatwith)) ENCODE_APP(nil, flatwith) -> c(APP(nil, flatwith)) ENCODE_APP(cons, flatwith) -> c(APP(cons, flatwith)) ENCODE_APP(flatwith, flatwith) -> c(APP(flatwith, flatwith)) ENCODE_APP(leaf, flatwith) -> c(APP(leaf, flatwith)) ENCODE_APP(node, flatwith) -> c(APP(node, flatwith)) ENCODE_APP(flatwithsub, flatwith) -> c(APP(flatwithsub, flatwith)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(node, flatwith) -> c(APP(node, flatwith)) ENCODE_APP(nil, flatwith) -> c(APP(nil, flatwith)) ENCODE_APP(flatwith, flatwith) -> c(APP(flatwith, flatwith)) ENCODE_APP(append, flatwith) -> c(APP(append, flatwith)) ENCODE_APP(leaf, flatwith) -> c(APP(leaf, flatwith)) ENCODE_APP(cons, flatwith) -> c(APP(cons, flatwith)) ENCODE_APP(flatwithsub, flatwith) -> c(APP(flatwithsub, flatwith)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, leaf) -> c(APP(encArg(x0), leaf)) by ENCODE_APP(append, leaf) -> c(APP(append, leaf)) ENCODE_APP(nil, leaf) -> c(APP(nil, leaf)) ENCODE_APP(cons, leaf) -> c(APP(cons, leaf)) ENCODE_APP(flatwith, leaf) -> c(APP(flatwith, leaf)) ENCODE_APP(leaf, leaf) -> c(APP(leaf, leaf)) ENCODE_APP(node, leaf) -> c(APP(node, leaf)) ENCODE_APP(flatwithsub, leaf) -> c(APP(flatwithsub, leaf)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(append, leaf) -> c(APP(append, leaf)) ENCODE_APP(nil, leaf) -> c(APP(nil, leaf)) ENCODE_APP(cons, leaf) -> c(APP(cons, leaf)) ENCODE_APP(flatwith, leaf) -> c(APP(flatwith, leaf)) ENCODE_APP(leaf, leaf) -> c(APP(leaf, leaf)) ENCODE_APP(node, leaf) -> c(APP(node, leaf)) ENCODE_APP(flatwithsub, leaf) -> c(APP(flatwithsub, leaf)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(flatwithsub, leaf) -> c(APP(flatwithsub, leaf)) ENCODE_APP(node, leaf) -> c(APP(node, leaf)) ENCODE_APP(leaf, leaf) -> c(APP(leaf, leaf)) ENCODE_APP(append, leaf) -> c(APP(append, leaf)) ENCODE_APP(nil, leaf) -> c(APP(nil, leaf)) ENCODE_APP(flatwith, leaf) -> c(APP(flatwith, leaf)) ENCODE_APP(cons, leaf) -> c(APP(cons, leaf)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, node) -> c(APP(encArg(x0), node)) by ENCODE_APP(append, node) -> c(APP(append, node)) ENCODE_APP(nil, node) -> c(APP(nil, node)) ENCODE_APP(cons, node) -> c(APP(cons, node)) ENCODE_APP(flatwith, node) -> c(APP(flatwith, node)) ENCODE_APP(leaf, node) -> c(APP(leaf, node)) ENCODE_APP(node, node) -> c(APP(node, node)) ENCODE_APP(flatwithsub, node) -> c(APP(flatwithsub, node)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(append, node) -> c(APP(append, node)) ENCODE_APP(nil, node) -> c(APP(nil, node)) ENCODE_APP(cons, node) -> c(APP(cons, node)) ENCODE_APP(flatwith, node) -> c(APP(flatwith, node)) ENCODE_APP(leaf, node) -> c(APP(leaf, node)) ENCODE_APP(node, node) -> c(APP(node, node)) ENCODE_APP(flatwithsub, node) -> c(APP(flatwithsub, node)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(nil, node) -> c(APP(nil, node)) ENCODE_APP(flatwith, node) -> c(APP(flatwith, node)) ENCODE_APP(flatwithsub, node) -> c(APP(flatwithsub, node)) ENCODE_APP(append, node) -> c(APP(append, node)) ENCODE_APP(node, node) -> c(APP(node, node)) ENCODE_APP(leaf, node) -> c(APP(leaf, node)) ENCODE_APP(cons, node) -> c(APP(cons, node)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_APP(x0, flatwithsub) -> c(APP(encArg(x0), flatwithsub)) by ENCODE_APP(append, flatwithsub) -> c(APP(append, flatwithsub)) ENCODE_APP(nil, flatwithsub) -> c(APP(nil, flatwithsub)) ENCODE_APP(cons, flatwithsub) -> c(APP(cons, flatwithsub)) ENCODE_APP(flatwith, flatwithsub) -> c(APP(flatwith, flatwithsub)) ENCODE_APP(leaf, flatwithsub) -> c(APP(leaf, flatwithsub)) ENCODE_APP(node, flatwithsub) -> c(APP(node, flatwithsub)) ENCODE_APP(flatwithsub, flatwithsub) -> c(APP(flatwithsub, flatwithsub)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(append, flatwithsub) -> c(APP(append, flatwithsub)) ENCODE_APP(nil, flatwithsub) -> c(APP(nil, flatwithsub)) ENCODE_APP(cons, flatwithsub) -> c(APP(cons, flatwithsub)) ENCODE_APP(flatwith, flatwithsub) -> c(APP(flatwith, flatwithsub)) ENCODE_APP(leaf, flatwithsub) -> c(APP(leaf, flatwithsub)) ENCODE_APP(node, flatwithsub) -> c(APP(node, flatwithsub)) ENCODE_APP(flatwithsub, flatwithsub) -> c(APP(flatwithsub, flatwithsub)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(leaf, flatwithsub) -> c(APP(leaf, flatwithsub)) ENCODE_APP(append, flatwithsub) -> c(APP(append, flatwithsub)) ENCODE_APP(flatwithsub, flatwithsub) -> c(APP(flatwithsub, flatwithsub)) ENCODE_APP(cons, flatwithsub) -> c(APP(cons, flatwithsub)) ENCODE_APP(flatwith, flatwithsub) -> c(APP(flatwith, flatwithsub)) ENCODE_APP(nil, flatwithsub) -> c(APP(nil, flatwithsub)) ENCODE_APP(node, flatwithsub) -> c(APP(node, flatwithsub)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(x0, cons_app(z0, z1)) -> c(APP(encArg(x0), app(encArg(z0), encArg(z1)))) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (127) 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, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(append, cons_app(x1, x2)) -> c(APP(append, 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(flatwith, cons_app(x1, x2)) -> c(APP(flatwith, app(encArg(x1), encArg(x2)))) ENCODE_APP(leaf, cons_app(x1, x2)) -> c(APP(leaf, app(encArg(x1), encArg(x2)))) ENCODE_APP(node, cons_app(x1, x2)) -> c(APP(node, app(encArg(x1), encArg(x2)))) ENCODE_APP(flatwithsub, cons_app(x1, x2)) -> c(APP(flatwithsub, 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)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(append, cons_app(x1, x2)) -> c(APP(append, 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(flatwith, cons_app(x1, x2)) -> c(APP(flatwith, app(encArg(x1), encArg(x2)))) ENCODE_APP(leaf, cons_app(x1, x2)) -> c(APP(leaf, app(encArg(x1), encArg(x2)))) ENCODE_APP(node, cons_app(x1, x2)) -> c(APP(node, app(encArg(x1), encArg(x2)))) ENCODE_APP(flatwithsub, cons_app(x1, x2)) -> c(APP(flatwithsub, 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)))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: ENCODE_APP(leaf, cons_app(x1, x2)) -> c(APP(leaf, app(encArg(x1), encArg(x2)))) ENCODE_APP(flatwith, cons_app(x1, x2)) -> c(APP(flatwith, app(encArg(x1), encArg(x2)))) ENCODE_APP(append, cons_app(x1, x2)) -> c(APP(append, app(encArg(x1), encArg(x2)))) ENCODE_APP(node, cons_app(x1, x2)) -> c(APP(node, app(encArg(x1), encArg(x2)))) ENCODE_APP(flatwithsub, cons_app(x1, x2)) -> c(APP(flatwithsub, 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)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCODE_APP(cons_app(z0, z1), x1) -> c(APP(app(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c_1, c7_3, c7_2, c21_1 ---------------------------------------- (131) 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), append) -> c(APP(app(encArg(x0), encArg(x1)), append)) 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), flatwith) -> c(APP(app(encArg(x0), encArg(x1)), flatwith)) ENCODE_APP(cons_app(x0, x1), leaf) -> c(APP(app(encArg(x0), encArg(x1)), leaf)) ENCODE_APP(cons_app(x0, x1), node) -> c(APP(app(encArg(x0), encArg(x1)), node)) ENCODE_APP(cons_app(x0, x1), flatwithsub) -> c(APP(app(encArg(x0), encArg(x1)), flatwithsub)) 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, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, 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(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(leaf, x1), x2) -> c(APP(app(leaf, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(node, x1), x2) -> c(APP(app(node, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, 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(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(leaf, x1), x2) -> c(APP(app(leaf, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(node, x1), x2) -> c(APP(app(node, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c7_3, c7_2, c21_1, c_1 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: 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(node, x1), x2) -> c(APP(app(node, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(leaf, x1), x2) -> c(APP(app(leaf, encArg(x1)), encArg(x2))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c17_1, c18_1, c19_1, c7_1, c21_3, c7_3, c7_2, c21_1, c_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(app(append, app(app(cons, z0), z1)), z2) -> c17(APP(app(append, z1), z2)) by APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c18_1, c19_1, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(app(flatwith, z0), app(leaf, z1)) -> c18(APP(z0, z1)) by APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c19_1, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1, c18_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace APP(app(flatwith, z0), app(node, z1)) -> c19(APP(app(flatwithsub, z0), z1)) by APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1, c18_1, c19_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_app(append, x1)) -> c7(ENCARG(x1)) by ENCARG(cons_app(append, cons_app(append, y0))) -> c7(ENCARG(cons_app(append, y0))) ENCARG(cons_app(append, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(append, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(append, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(append, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(append, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(append, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) ENCARG(cons_app(append, cons_app(append, y0))) -> c7(ENCARG(cons_app(append, y0))) ENCARG(cons_app(append, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(append, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(append, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(append, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(append, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(append, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1, c18_1, c19_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(leaf, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x3))) by APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(append, nil), app(app(flatwithsub, z0), z3)))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) ENCARG(cons_app(append, cons_app(append, y0))) -> c7(ENCARG(cons_app(append, y0))) ENCARG(cons_app(append, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(append, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(append, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(append, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(append, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(append, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(append, nil), app(app(flatwithsub, z0), z3)))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(append, nil), app(app(flatwithsub, z0), z3)))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1, c18_1, c19_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_app(nil, x1)) -> c7(ENCARG(x1)) by ENCARG(cons_app(nil, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(nil, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(nil, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(nil, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(nil, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(nil, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(nil, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(nil, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(nil, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) ENCARG(cons_app(nil, cons_app(append, cons_app(append, y0)))) -> c7(ENCARG(cons_app(append, cons_app(append, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(nil, y0)))) -> c7(ENCARG(cons_app(append, cons_app(nil, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons, y0)))) -> c7(ENCARG(cons_app(append, cons_app(cons, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwith, y0)))) -> c7(ENCARG(cons_app(append, cons_app(flatwith, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(leaf, y0)))) -> c7(ENCARG(cons_app(append, cons_app(leaf, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(node, y0)))) -> c7(ENCARG(cons_app(append, cons_app(node, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwithsub, y0)))) -> c7(ENCARG(cons_app(append, cons_app(flatwithsub, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, append))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, nil))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, cons))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, flatwith))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, leaf))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, node))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, flatwithsub))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3)))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3)))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(append, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(nil, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(cons, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(flatwith, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(leaf, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(node, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(flatwithsub, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3))))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3))))) ENCARG(cons_app(nil, cons_app(append, cons_app(append, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(nil, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwith, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(leaf, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(node, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwithsub, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, append), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, nil), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, cons), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, flatwith), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, leaf), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, node), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, flatwithsub), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(append, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(flatwith, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(flatwithsub, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), append)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), nil)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), cons)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), flatwith)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), leaf)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), node)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), flatwithsub)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(nil, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(cons, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(leaf, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(node, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) ENCARG(cons_app(append, cons_app(append, y0))) -> c7(ENCARG(cons_app(append, y0))) ENCARG(cons_app(append, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(append, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(append, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(append, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(append, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(append, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(append, nil), app(app(flatwithsub, z0), z3)))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) ENCARG(cons_app(nil, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(nil, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(nil, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(nil, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(nil, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(nil, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(nil, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(nil, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(nil, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) ENCARG(cons_app(nil, cons_app(append, cons_app(append, y0)))) -> c7(ENCARG(cons_app(append, cons_app(append, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(nil, y0)))) -> c7(ENCARG(cons_app(append, cons_app(nil, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons, y0)))) -> c7(ENCARG(cons_app(append, cons_app(cons, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwith, y0)))) -> c7(ENCARG(cons_app(append, cons_app(flatwith, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(leaf, y0)))) -> c7(ENCARG(cons_app(append, cons_app(leaf, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(node, y0)))) -> c7(ENCARG(cons_app(append, cons_app(node, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwithsub, y0)))) -> c7(ENCARG(cons_app(append, cons_app(flatwithsub, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, append))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, nil))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, cons))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, flatwith))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, leaf))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, node))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, flatwithsub))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3)))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3)))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(append, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(nil, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(cons, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(flatwith, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(leaf, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(node, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(flatwithsub, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3))))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3))))) ENCARG(cons_app(nil, cons_app(append, cons_app(append, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(nil, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwith, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(leaf, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(node, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwithsub, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, append), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, nil), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, cons), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, flatwith), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, leaf), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, node), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, flatwithsub), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(append, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(flatwith, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(flatwithsub, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), append)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), nil)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), cons)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), flatwith)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), leaf)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), node)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), flatwithsub)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(nil, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(cons, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(leaf, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(node, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1)))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(append, nil), app(app(flatwithsub, z0), z3)))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1, c18_1, c19_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(append, nil), app(app(flatwithsub, z0), z3)))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) by APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(flatwithsub, z0), z3))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: encArg(append) -> append encArg(nil) -> nil encArg(cons) -> cons encArg(flatwith) -> flatwith encArg(leaf) -> leaf encArg(node) -> node encArg(flatwithsub) -> flatwithsub encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1)) app(app(append, nil), z0) -> z0 app(app(append, app(app(cons, z0), z1)), z2) -> app(app(cons, z0), app(app(append, z1), z2)) app(app(flatwith, z0), app(leaf, z1)) -> app(app(cons, app(z0, z1)), nil) app(app(flatwith, z0), app(node, z1)) -> app(app(flatwithsub, z0), z1) app(app(flatwithsub, z0), nil) -> nil app(app(flatwithsub, z0), app(app(cons, z1), z2)) -> app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)) Tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 ENCARG(cons_app(cons, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwith, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(leaf, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(node, x1)) -> c7(ENCARG(x1)) ENCARG(cons_app(flatwithsub, x1)) -> c7(ENCARG(x1)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) ENCARG(cons_app(x0, cons_app(x1, append))) -> c7(APP(encArg(x0), app(encArg(x1), append)), ENCARG(x0), ENCARG(cons_app(x1, append))) ENCARG(cons_app(x0, cons_app(x1, nil))) -> c7(APP(encArg(x0), app(encArg(x1), nil)), ENCARG(x0), ENCARG(cons_app(x1, nil))) ENCARG(cons_app(x0, cons_app(x1, cons))) -> c7(APP(encArg(x0), app(encArg(x1), cons)), ENCARG(x0), ENCARG(cons_app(x1, cons))) ENCARG(cons_app(x0, cons_app(x1, flatwith))) -> c7(APP(encArg(x0), app(encArg(x1), flatwith)), ENCARG(x0), ENCARG(cons_app(x1, flatwith))) ENCARG(cons_app(x0, cons_app(x1, leaf))) -> c7(APP(encArg(x0), app(encArg(x1), leaf)), ENCARG(x0), ENCARG(cons_app(x1, leaf))) ENCARG(cons_app(x0, cons_app(x1, node))) -> c7(APP(encArg(x0), app(encArg(x1), node)), ENCARG(x0), ENCARG(cons_app(x1, node))) ENCARG(cons_app(x0, cons_app(x1, flatwithsub))) -> c7(APP(encArg(x0), app(encArg(x1), flatwithsub)), ENCARG(x0), ENCARG(cons_app(x1, flatwithsub))) ENCARG(cons_app(x0, cons_app(x1, cons_app(z0, z1)))) -> c7(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(append, x2))) -> c7(APP(encArg(x0), app(append, encArg(x2))), ENCARG(x0), ENCARG(cons_app(append, x2))) ENCARG(cons_app(x0, cons_app(nil, x2))) -> c7(APP(encArg(x0), app(nil, encArg(x2))), ENCARG(x0), ENCARG(cons_app(nil, x2))) ENCARG(cons_app(x0, cons_app(cons, x2))) -> c7(APP(encArg(x0), app(cons, encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons, x2))) ENCARG(cons_app(x0, cons_app(flatwith, x2))) -> c7(APP(encArg(x0), app(flatwith, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwith, x2))) ENCARG(cons_app(x0, cons_app(leaf, x2))) -> c7(APP(encArg(x0), app(leaf, encArg(x2))), ENCARG(x0), ENCARG(cons_app(leaf, x2))) ENCARG(cons_app(x0, cons_app(node, x2))) -> c7(APP(encArg(x0), app(node, encArg(x2))), ENCARG(x0), ENCARG(cons_app(node, x2))) ENCARG(cons_app(x0, cons_app(flatwithsub, x2))) -> c7(APP(encArg(x0), app(flatwithsub, encArg(x2))), ENCARG(x0), ENCARG(cons_app(flatwithsub, x2))) ENCARG(cons_app(x0, cons_app(cons_app(z0, z1), x2))) -> c7(APP(encArg(x0), app(app(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_app(cons_app(z0, z1), x2))) ENCARG(cons_app(cons_app(z0, z1), cons_app(x1, x2))) -> c7(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2))), ENCARG(cons_app(z0, z1)), ENCARG(cons_app(x1, x2))) ENCARG(cons_app(append, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(nil, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwith, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(leaf, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(node, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(flatwithsub, cons_app(x1, x2))) -> c7(ENCARG(cons_app(x1, x2))) ENCARG(cons_app(cons_app(x0, append), x2)) -> c7(APP(app(encArg(x0), append), encArg(x2)), ENCARG(cons_app(x0, append)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, nil), x2)) -> c7(APP(app(encArg(x0), nil), encArg(x2)), ENCARG(cons_app(x0, nil)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons), x2)) -> c7(APP(app(encArg(x0), cons), encArg(x2)), ENCARG(cons_app(x0, cons)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwith), x2)) -> c7(APP(app(encArg(x0), flatwith), encArg(x2)), ENCARG(cons_app(x0, flatwith)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, leaf), x2)) -> c7(APP(app(encArg(x0), leaf), encArg(x2)), ENCARG(cons_app(x0, leaf)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, node), x2)) -> c7(APP(app(encArg(x0), node), encArg(x2)), ENCARG(cons_app(x0, node)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, flatwithsub), x2)) -> c7(APP(app(encArg(x0), flatwithsub), encArg(x2)), ENCARG(cons_app(x0, flatwithsub)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, cons_app(z0, z1)), x2)) -> c7(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(append, x1), x2)) -> c7(APP(app(append, encArg(x1)), encArg(x2)), ENCARG(cons_app(append, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwith, x1), x2)) -> c7(APP(app(flatwith, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwith, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(flatwithsub, x1), x2)) -> c7(APP(app(flatwithsub, encArg(x1)), encArg(x2)), ENCARG(cons_app(flatwithsub, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons_app(z0, z1), x1), x2)) -> c7(APP(app(app(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_app(cons_app(z0, z1), x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(x0, x1), append)) -> c7(APP(app(encArg(x0), encArg(x1)), append), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), nil)) -> c7(APP(app(encArg(x0), encArg(x1)), nil), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), cons)) -> c7(APP(app(encArg(x0), encArg(x1)), cons), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwith)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwith), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), leaf)) -> c7(APP(app(encArg(x0), encArg(x1)), leaf), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), node)) -> c7(APP(app(encArg(x0), encArg(x1)), node), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(x0, x1), flatwithsub)) -> c7(APP(app(encArg(x0), encArg(x1)), flatwithsub), ENCARG(cons_app(x0, x1))) ENCARG(cons_app(cons_app(nil, x1), x2)) -> c7(ENCARG(cons_app(nil, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(cons, x1), x2)) -> c7(ENCARG(cons_app(cons, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(leaf, x1), x2)) -> c7(ENCARG(cons_app(leaf, x1)), ENCARG(x2)) ENCARG(cons_app(cons_app(node, x1), x2)) -> c7(ENCARG(cons_app(node, x1)), ENCARG(x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) ENCODE_APP(cons_app(z0, z1), append) -> c(APP(app(encArg(z0), encArg(z1)), append)) ENCODE_APP(cons_app(z0, z1), nil) -> c(APP(app(encArg(z0), encArg(z1)), nil)) ENCODE_APP(cons_app(z0, z1), cons) -> c(APP(app(encArg(z0), encArg(z1)), cons)) ENCODE_APP(cons_app(z0, z1), flatwith) -> c(APP(app(encArg(z0), encArg(z1)), flatwith)) ENCODE_APP(cons_app(z0, z1), leaf) -> c(APP(app(encArg(z0), encArg(z1)), leaf)) ENCODE_APP(cons_app(z0, z1), node) -> c(APP(app(encArg(z0), encArg(z1)), node)) ENCODE_APP(cons_app(z0, z1), flatwithsub) -> c(APP(app(encArg(z0), encArg(z1)), flatwithsub)) ENCODE_APP(x0, cons_app(x1, append)) -> c(APP(encArg(x0), app(encArg(x1), append))) 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, flatwith)) -> c(APP(encArg(x0), app(encArg(x1), flatwith))) ENCODE_APP(x0, cons_app(x1, leaf)) -> c(APP(encArg(x0), app(encArg(x1), leaf))) ENCODE_APP(x0, cons_app(x1, node)) -> c(APP(encArg(x0), app(encArg(x1), node))) ENCODE_APP(x0, cons_app(x1, flatwithsub)) -> c(APP(encArg(x0), app(encArg(x1), flatwithsub))) 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(append, x2)) -> c(APP(encArg(x0), app(append, 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(flatwith, x2)) -> c(APP(encArg(x0), app(flatwith, encArg(x2)))) ENCODE_APP(x0, cons_app(leaf, x2)) -> c(APP(encArg(x0), app(leaf, encArg(x2)))) ENCODE_APP(x0, cons_app(node, x2)) -> c(APP(encArg(x0), app(node, encArg(x2)))) ENCODE_APP(x0, cons_app(flatwithsub, x2)) -> c(APP(encArg(x0), app(flatwithsub, 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(cons_app(z0, z1), cons_app(x1, x2)) -> c(APP(app(encArg(z0), encArg(z1)), app(encArg(x1), encArg(x2)))) ENCODE_APP(cons_app(x0, append), x2) -> c(APP(app(encArg(x0), append), 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, flatwith), x2) -> c(APP(app(encArg(x0), flatwith), encArg(x2))) ENCODE_APP(cons_app(x0, leaf), x2) -> c(APP(app(encArg(x0), leaf), encArg(x2))) ENCODE_APP(cons_app(x0, node), x2) -> c(APP(app(encArg(x0), node), encArg(x2))) ENCODE_APP(cons_app(x0, flatwithsub), x2) -> c(APP(app(encArg(x0), flatwithsub), 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(append, x1), x2) -> c(APP(app(append, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwith, x1), x2) -> c(APP(app(flatwith, encArg(x1)), encArg(x2))) ENCODE_APP(cons_app(flatwithsub, x1), x2) -> c(APP(app(flatwithsub, 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))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) ENCARG(cons_app(append, cons_app(append, y0))) -> c7(ENCARG(cons_app(append, y0))) ENCARG(cons_app(append, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(append, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(append, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(append, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(append, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(append, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) ENCARG(cons_app(nil, cons_app(nil, y0))) -> c7(ENCARG(cons_app(nil, y0))) ENCARG(cons_app(nil, cons_app(cons, y0))) -> c7(ENCARG(cons_app(cons, y0))) ENCARG(cons_app(nil, cons_app(flatwith, y0))) -> c7(ENCARG(cons_app(flatwith, y0))) ENCARG(cons_app(nil, cons_app(leaf, y0))) -> c7(ENCARG(cons_app(leaf, y0))) ENCARG(cons_app(nil, cons_app(node, y0))) -> c7(ENCARG(cons_app(node, y0))) ENCARG(cons_app(nil, cons_app(flatwithsub, y0))) -> c7(ENCARG(cons_app(flatwithsub, y0))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, append)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, append)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, nil)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, nil)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, cons)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, flatwith)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwith)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, leaf)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, leaf)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, node)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, node)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, flatwithsub)))) -> c7(ENCARG(cons_app(y0, cons_app(y1, flatwithsub)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(y1, cons_app(y2, y3))))) -> c7(ENCARG(cons_app(y0, cons_app(y1, cons_app(y2, y3))))) ENCARG(cons_app(nil, cons_app(y0, cons_app(append, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(append, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(nil, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(nil, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(cons, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(cons, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(flatwith, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwith, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(leaf, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(leaf, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(node, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(node, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(flatwithsub, y1)))) -> c7(ENCARG(cons_app(y0, cons_app(flatwithsub, y1)))) ENCARG(cons_app(nil, cons_app(y0, cons_app(cons_app(y1, y2), y3)))) -> c7(ENCARG(cons_app(y0, cons_app(cons_app(y1, y2), y3)))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), cons_app(y2, y3)))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons_app(y2, y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(append, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(nil, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(nil, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(cons, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(cons, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(flatwith, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwith, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(leaf, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(leaf, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(node, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(node, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(flatwithsub, cons_app(y0, y1)))) -> c7(ENCARG(cons_app(flatwithsub, cons_app(y0, y1)))) ENCARG(cons_app(nil, cons_app(cons_app(y0, append), y1))) -> c7(ENCARG(cons_app(cons_app(y0, append), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, nil), y1))) -> c7(ENCARG(cons_app(cons_app(y0, nil), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, cons), y1))) -> c7(ENCARG(cons_app(cons_app(y0, cons), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, flatwith), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwith), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, leaf), y1))) -> c7(ENCARG(cons_app(cons_app(y0, leaf), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, node), y1))) -> c7(ENCARG(cons_app(cons_app(y0, node), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, flatwithsub), y1))) -> c7(ENCARG(cons_app(cons_app(y0, flatwithsub), y1))) ENCARG(cons_app(nil, cons_app(cons_app(y0, cons_app(y1, y2)), y3))) -> c7(ENCARG(cons_app(cons_app(y0, cons_app(y1, y2)), y3))) ENCARG(cons_app(nil, cons_app(cons_app(append, y0), y1))) -> c7(ENCARG(cons_app(cons_app(append, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(flatwith, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwith, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(flatwithsub, y0), y1))) -> c7(ENCARG(cons_app(cons_app(flatwithsub, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(cons_app(y0, y1), y2), y3))) -> c7(ENCARG(cons_app(cons_app(cons_app(y0, y1), y2), y3))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), append))) -> c7(ENCARG(cons_app(cons_app(y0, y1), append))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), nil))) -> c7(ENCARG(cons_app(cons_app(y0, y1), nil))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), cons))) -> c7(ENCARG(cons_app(cons_app(y0, y1), cons))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), flatwith))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwith))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), leaf))) -> c7(ENCARG(cons_app(cons_app(y0, y1), leaf))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), node))) -> c7(ENCARG(cons_app(cons_app(y0, y1), node))) ENCARG(cons_app(nil, cons_app(cons_app(y0, y1), flatwithsub))) -> c7(ENCARG(cons_app(cons_app(y0, y1), flatwithsub))) ENCARG(cons_app(nil, cons_app(cons_app(nil, y0), y1))) -> c7(ENCARG(cons_app(cons_app(nil, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(cons, y0), y1))) -> c7(ENCARG(cons_app(cons_app(cons, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(leaf, y0), y1))) -> c7(ENCARG(cons_app(cons_app(leaf, y0), y1))) ENCARG(cons_app(nil, cons_app(cons_app(node, y0), y1))) -> c7(ENCARG(cons_app(cons_app(node, y0), y1))) ENCARG(cons_app(nil, cons_app(append, cons_app(append, y0)))) -> c7(ENCARG(cons_app(append, cons_app(append, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(nil, y0)))) -> c7(ENCARG(cons_app(append, cons_app(nil, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons, y0)))) -> c7(ENCARG(cons_app(append, cons_app(cons, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwith, y0)))) -> c7(ENCARG(cons_app(append, cons_app(flatwith, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(leaf, y0)))) -> c7(ENCARG(cons_app(append, cons_app(leaf, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(node, y0)))) -> c7(ENCARG(cons_app(append, cons_app(node, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwithsub, y0)))) -> c7(ENCARG(cons_app(append, cons_app(flatwithsub, y0)))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, append))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, append))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, nil))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, nil))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, cons))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, flatwith))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwith))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, leaf))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, leaf))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, node))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, node))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, flatwithsub))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, flatwithsub))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3)))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(y1, cons_app(y2, y3)))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(append, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(append, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(nil, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(nil, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(cons, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(cons, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(flatwith, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(flatwith, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(leaf, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(leaf, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(node, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(node, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(flatwithsub, y1))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(flatwithsub, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3))))) -> c7(ENCARG(cons_app(append, cons_app(y0, cons_app(cons_app(y1, y2), y3))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3))))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons_app(y2, y3))))) ENCARG(cons_app(nil, cons_app(append, cons_app(append, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(append, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(nil, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(nil, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(cons, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwith, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(flatwith, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(leaf, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(leaf, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(node, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(node, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(flatwithsub, cons_app(y0, y1))))) -> c7(ENCARG(cons_app(append, cons_app(flatwithsub, cons_app(y0, y1))))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, append), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, append), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, nil), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, nil), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, cons), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, cons), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, flatwith), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, flatwith), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, leaf), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, leaf), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, node), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, node), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, flatwithsub), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, flatwithsub), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, cons_app(y1, y2)), y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(append, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(append, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(flatwith, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(flatwith, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(flatwithsub, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(flatwithsub, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(cons_app(y0, y1), y2), y3)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), append)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), append)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), nil)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), nil)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), cons)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), cons)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), flatwith)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwith)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), leaf)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), leaf)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), node)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), node)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(y0, y1), flatwithsub)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(y0, y1), flatwithsub)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(nil, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(nil, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(cons, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(cons, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(leaf, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(leaf, y0), y1)))) ENCARG(cons_app(nil, cons_app(append, cons_app(cons_app(node, y0), y1)))) -> c7(ENCARG(cons_app(append, cons_app(cons_app(node, y0), y1)))) APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(flatwithsub, z0), z3))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) S tuples: APP(app(append, nil), z0) -> c16 APP(app(flatwithsub, z0), nil) -> c20 APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), x2)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), nil)) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), nil), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), nil)) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), nil), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), nil)) APP(app(flatwithsub, x0), app(app(cons, x1), nil)) -> c21(APP(app(flatwithsub, x0), nil)) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), nil))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), nil)), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), nil))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, x2), app(app(cons, z1), z2)))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwith, z0), x2)), app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2)))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, x2), app(app(cons, z1), z2)))) APP(app(flatwithsub, z0), app(app(cons, x1), app(app(cons, app(node, z1)), x3))) -> c21(APP(app(append, app(app(flatwith, z0), x1)), app(app(append, app(app(flatwithsub, z0), z1)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), x1), APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), x3))) APP(app(flatwithsub, z0), app(app(cons, app(leaf, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(cons, app(z0, z1)), nil)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(leaf, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, z1)), app(app(cons, x2), x3))) -> c21(APP(app(append, app(app(flatwithsub, z0), z1)), app(app(append, app(app(flatwith, z0), x2)), app(app(flatwithsub, z0), x3))), APP(app(flatwith, z0), app(node, z1)), APP(app(flatwithsub, z0), app(app(cons, x2), x3))) APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), x2)) -> c21(APP(app(append, nil), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, nil)), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, z1), z2))), x2)) -> c21(APP(app(append, app(app(append, app(app(flatwith, z0), z1)), app(app(flatwithsub, z0), z2))), app(app(flatwithsub, z0), x2)), APP(app(flatwith, z0), app(node, app(app(cons, z1), z2))), APP(app(flatwithsub, z0), x2)) APP(app(flatwithsub, x0), app(app(cons, app(node, x1)), x2)) -> c21(APP(app(flatwith, x0), app(node, x1))) APP(app(append, app(app(cons, z0), nil)), z2) -> c17(APP(app(append, nil), z2)) APP(app(append, app(app(cons, z0), app(app(cons, y0), y1))), z2) -> c17(APP(app(append, app(app(cons, y0), y1)), z2)) APP(app(flatwith, app(append, nil)), app(leaf, z1)) -> c18(APP(app(append, nil), z1)) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, nil)) -> c18(APP(app(flatwithsub, y0), nil)) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(leaf, y1))) -> c18(APP(app(flatwith, y0), app(leaf, y1))) APP(app(flatwith, app(flatwith, y0)), app(leaf, app(node, y1))) -> c18(APP(app(flatwith, y0), app(node, y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), nil))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), nil))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), nil)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, nil)), y1))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, app(flatwithsub, y0)), app(leaf, app(app(cons, app(node, y1)), y2))) -> c18(APP(app(flatwithsub, y0), app(app(cons, app(node, y1)), y2))) APP(app(flatwith, app(append, app(app(cons, y0), nil))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), nil)), z1)) APP(app(flatwith, app(append, app(app(cons, y0), app(app(cons, y1), y2)))), app(leaf, z1)) -> c18(APP(app(append, app(app(cons, y0), app(app(cons, y1), y2))), z1)) APP(app(flatwith, z0), app(node, nil)) -> c19(APP(app(flatwithsub, z0), nil)) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), y2))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), nil))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), nil))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), nil)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), nil)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, y2), app(app(cons, y3), y4))))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(leaf, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, y1), app(app(cons, app(node, y2)), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(leaf, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), app(app(cons, y2), y3)))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, nil)), y1))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, nil)), y1))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, app(app(cons, y1), y2))), y3))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, app(app(cons, y1), y2))), y3))) APP(app(flatwith, z0), app(node, app(app(cons, app(node, y1)), y2))) -> c19(APP(app(flatwithsub, z0), app(app(cons, app(node, y1)), y2))) APP(app(flatwithsub, z0), app(app(cons, z1), app(app(cons, app(leaf, z2)), z3))) -> c21(APP(app(append, app(app(flatwith, z0), z1)), app(app(cons, app(z0, z2)), app(app(flatwithsub, z0), z3))), APP(app(flatwith, z0), z1), APP(app(flatwithsub, z0), app(app(cons, app(leaf, z2)), z3))) K tuples:none Defined Rule Symbols: encArg_1, app_2 Defined Pair Symbols: APP_2, ENCARG_1, ENCODE_APP_2 Compound Symbols: c16, c20, c7_1, c21_3, c7_3, c7_2, c21_1, c_1, c17_1, c18_1, c19_1