KILLED proof of input_FWnJWm6MK6.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 13 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 50 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 321 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 150 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 247 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 366 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 129 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 20.7 s] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 16.1 s] (56) CpxRNTS (57) CompletionProof [UPPER BOUND(ID), 0 ms] (58) CpxTypedWeightedCompleteTrs (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms] (96) CdtProblem (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (106) CdtProblem (107) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (140) CdtProblem (141) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (144) CdtProblem (145) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y and(true, y) -> y and(false, y) -> false size(empty) -> 0 size(edge(x, y, i)) -> s(size(i)) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) reachable(x, y, i) -> reach(x, y, 0, i, i) reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) if1(true, x, y, c, i, j) -> true if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) if2(false, x, y, c, i, j) -> false if2(true, x, y, c, empty, j) -> false if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0', 0') -> true eq(0', s(x)) -> false eq(s(x), 0') -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y and(true, y) -> y and(false, y) -> false size(empty) -> 0' size(edge(x, y, i)) -> s(size(i)) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) reachable(x, y, i) -> reach(x, y, 0', i, i) reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) if1(true, x, y, c, i, j) -> true if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) if2(false, x, y, c, i, j) -> false if2(true, x, y, c, empty, j) -> false if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y and(true, y) -> y and(false, y) -> false size(empty) -> 0 size(edge(x, y, i)) -> s(size(i)) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) reachable(x, y, i) -> reach(x, y, 0, i, i) reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) if1(true, x, y, c, i, j) -> true if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) if2(false, x, y, c, i, j) -> false if2(true, x, y, c, empty, j) -> false if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) 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: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] or(true, y) -> true [1] or(false, y) -> y [1] and(true, y) -> y [1] and(false, y) -> false [1] size(empty) -> 0 [1] size(edge(x, y, i)) -> s(size(i)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] reachable(x, y, i) -> reach(x, y, 0, i, i) [1] reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1] if1(true, x, y, c, i, j) -> true [1] if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1] if2(false, x, y, c, i, j) -> false [1] if2(true, x, y, c, empty, j) -> false [1] if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] or(true, y) -> true [1] or(false, y) -> y [1] and(true, y) -> y [1] and(false, y) -> false [1] size(empty) -> 0 [1] size(edge(x, y, i)) -> s(size(i)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] reachable(x, y, i) -> reach(x, y, 0, i, i) [1] reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1] if1(true, x, y, c, i, j) -> true [1] if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1] if2(false, x, y, c, i, j) -> false [1] if2(true, x, y, c, empty, j) -> false [1] if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false or :: true:false -> true:false -> true:false and :: true:false -> true:false -> true:false size :: empty:edge -> 0:s empty :: empty:edge edge :: 0:s -> 0:s -> empty:edge -> empty:edge le :: 0:s -> 0:s -> true:false reachable :: 0:s -> 0:s -> empty:edge -> true:false reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: reachable_3 (c) The following functions are completely defined: le_2 size_1 eq_2 if2_6 and_2 reach_5 if1_6 or_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) 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: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] or(true, y) -> true [1] or(false, y) -> y [1] and(true, y) -> y [1] and(false, y) -> false [1] size(empty) -> 0 [1] size(edge(x, y, i)) -> s(size(i)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] reachable(x, y, i) -> reach(x, y, 0, i, i) [1] reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1] if1(true, x, y, c, i, j) -> true [1] if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1] if2(false, x, y, c, i, j) -> false [1] if2(true, x, y, c, empty, j) -> false [1] if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false or :: true:false -> true:false -> true:false and :: true:false -> true:false -> true:false size :: empty:edge -> 0:s empty :: empty:edge edge :: 0:s -> 0:s -> empty:edge -> empty:edge le :: 0:s -> 0:s -> true:false reachable :: 0:s -> 0:s -> empty:edge -> true:false reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) 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: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] or(true, y) -> true [1] or(false, y) -> y [1] and(true, y) -> y [1] and(false, y) -> false [1] size(empty) -> 0 [1] size(edge(x, y, i)) -> s(size(i)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] reachable(x, y, i) -> reach(x, y, 0, i, i) [1] reach(0, 0, c, i, j) -> if1(true, 0, 0, c, i, j) [2] reach(0, s(x'), c, i, j) -> if1(false, 0, s(x'), c, i, j) [2] reach(s(x''), 0, c, i, j) -> if1(false, s(x''), 0, c, i, j) [2] reach(s(x1), s(y'), c, i, j) -> if1(eq(x1, y'), s(x1), s(y'), c, i, j) [2] if1(true, x, y, c, i, j) -> true [1] if1(false, x, y, c, i, empty) -> if2(le(c, 0), x, y, c, i, empty) [2] if1(false, x, y, c, i, edge(x2, y'', i')) -> if2(le(c, s(size(i'))), x, y, c, i, edge(x2, y'', i')) [2] if2(false, x, y, c, i, j) -> false [1] if2(true, x, y, c, empty, j) -> false [1] if2(true, 0, y, c, edge(0, v, empty), j) -> or(false, and(true, if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, 0, y, c, edge(s(x3), v, empty), j) -> or(false, and(false, if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, s(x4), y, c, edge(0, v, empty), j) -> or(false, and(false, if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, s(x5), y, c, edge(s(y1), v, empty), j) -> or(false, and(eq(x5, y1), if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, 0, y, c, edge(0, v, edge(u', v', i'')), j) -> or(or(if2(true, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, s(c), j, j))), and(true, if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, 0, y, c, edge(s(x6), v, edge(u', v', i'')), j) -> or(or(if2(true, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, s(c), j, j))), and(false, if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, s(x7), y, c, edge(0, v, edge(u', v', i'')), j) -> or(or(if2(true, s(x7), y, c, i'', j), and(eq(s(x7), u'), reach(v', y, s(c), j, j))), and(false, if1(eq(v, y), v, y, s(c), j, j))) [4] if2(true, s(x8), y, c, edge(s(y2), v, edge(u', v', i'')), j) -> or(or(if2(true, s(x8), y, c, i'', j), and(eq(s(x8), u'), reach(v', y, s(c), j, j))), and(eq(x8, y2), if1(eq(v, y), v, y, s(c), j, j))) [4] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false or :: true:false -> true:false -> true:false and :: true:false -> true:false -> true:false size :: empty:edge -> 0:s empty :: empty:edge edge :: 0:s -> 0:s -> empty:edge -> empty:edge le :: 0:s -> 0:s -> true:false reachable :: 0:s -> 0:s -> empty:edge -> true:false reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 empty => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 0), x, y, c, i, 0) :|: z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z3 = 0, x >= 0, y >= 0, i >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 1 + size(i')), x, y, c, i, 1 + x2 + y'' + i') :|: z' = x, z'' = y, z1 = c, c >= 0, y >= 0, i >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z2 = i, x >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z3 = j, i >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, j >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x7, y, c, i'', j), and(eq(1 + x7, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: z' = 1 + x7, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x7 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x8, y, c, i'', j), and(eq(1 + x8, u'), reach(v', y, 1 + c, j, j))), and(eq(x8, y2), if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, x8 >= 0, z'' = y, z1 = c, c >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), y >= 0, z3 = j, u' >= 0, y2 >= 0, j >= 0, i'' >= 0, z' = 1 + x8, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(x5, y1), if1(eq(v, y), v, y, 1 + c, j, j))) :|: y1 >= 0, j >= 0, x5 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z' = 1 + x5, z3 = j, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, z2 = 1 + 0 + v + 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, x4 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, z' = 1 + x4, y >= 0, z3 = j, z2 = 1 + 0 + v + 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, z2 = 0, x >= 0, y >= 0, z3 = j le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(x1, y'), 1 + x1, 1 + y', c, i, j) :|: j >= 0, x1 >= 0, c >= 0, z2 = j, i >= 0, z = 1 + x1, y' >= 0, z' = 1 + y', z'' = c, z1 = i reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, c, i, j) :|: j >= 0, c >= 0, z2 = j, i >= 0, z = 0, z'' = c, z1 = i, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + x', c, i, j) :|: j >= 0, z' = 1 + x', c >= 0, z2 = j, x' >= 0, i >= 0, z = 0, z'' = c, z1 = i reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + x'', 0, c, i, j) :|: z = 1 + x'', j >= 0, c >= 0, z2 = j, i >= 0, x'' >= 0, z'' = c, z1 = i, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(x, y, 0, i, i) :|: z'' = i, x >= 0, y >= 0, i >= 0, z = x, z' = y size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 0), x, y, c, i, 0) :|: z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z3 = 0, x >= 0, y >= 0, i >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(c, 1 + size(i')), x, y, c, i, 1 + x2 + y'' + i') :|: z' = x, z'' = y, z1 = c, c >= 0, y >= 0, i >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z2 = i, x >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z3 = j, i >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, j >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, y, c, i'', j), and(eq(0, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x7, y, c, i'', j), and(eq(1 + x7, u'), reach(v', y, 1 + c, j, j))), and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: z' = 1 + x7, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), j >= 0, i'' >= 0, x7 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + x8, y, c, i'', j), and(eq(1 + x8, u'), reach(v', y, 1 + c, j, j))), and(eq(x8, y2), if1(eq(v, y), v, y, 1 + c, j, j))) :|: v >= 0, x8 >= 0, z'' = y, z1 = c, c >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), y >= 0, z3 = j, u' >= 0, y2 >= 0, j >= 0, i'' >= 0, z' = 1 + x8, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(x5, y1), if1(eq(v, y), v, y, 1 + c, j, j))) :|: y1 >= 0, j >= 0, x5 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z' = 1 + x5, z3 = j, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, z2 = 1 + 0 + v + 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, y), v, y, 1 + c, j, j))) :|: j >= 0, x4 >= 0, v >= 0, z'' = y, z1 = c, c >= 0, z = 1, z' = 1 + x4, y >= 0, z3 = j, z2 = 1 + 0 + v + 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, z2 = 0, x >= 0, y >= 0, z3 = j le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(x1, y'), 1 + x1, 1 + y', c, i, j) :|: j >= 0, x1 >= 0, c >= 0, z2 = j, i >= 0, z = 1 + x1, y' >= 0, z' = 1 + y', z'' = c, z1 = i reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, c, i, j) :|: j >= 0, c >= 0, z2 = j, i >= 0, z = 0, z'' = c, z1 = i, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + x', c, i, j) :|: j >= 0, z' = 1 + x', c >= 0, z2 = j, x' >= 0, i >= 0, z = 0, z'' = c, z1 = i reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + x'', 0, c, i, j) :|: z = 1 + x'', j >= 0, c >= 0, z2 = j, i >= 0, x'' >= 0, z'' = c, z1 = i, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(x, y, 0, i, i) :|: z'' = i, x >= 0, y >= 0, i >= 0, z = x, z' = y size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { size } { eq } { and } { or } { if2, reach, if1 } { reachable } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {le}, {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {le}, {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {le}, {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 0), z', z'', z1, z2, 0) :|: z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: size after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {size}, {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: size after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 2 }-> if2(le(z1, 1 + size(i')), z', z'', z1, z2, 1 + x2 + y'' + i') :|: z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {eq}, {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(1, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 0, z'', z1, i'', z3), and(eq(0, u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(eq(z' - 1, y2), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(eq(1 + (z' - 1), u'), reach(v', z'', 1 + z1, z3, z3))), and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(eq(z' - 1, y1), if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(1, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(v, z''), v, z'', 1 + z1, z3, z3))) :|: z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 4 }-> or(0, and(0, if1(eq(z2 - 1, z''), z2 - 1, z'', 1 + z1, z3, z3))) :|: z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(eq(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {and}, {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {or}, {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] or: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] or: runtime: O(1) [1], size: O(n^1) [1 + z'] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] or: runtime: O(1) [1], size: O(n^1) [1 + z'] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: reach after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] or: runtime: O(1) [1], size: O(n^1) [1 + z'] if2: runtime: ?, size: INF reach: runtime: ?, size: INF if1: runtime: ?, size: INF ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 eq(z, z') -{ 3 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 4 }-> if2(s', z', z'', z1, z2, 0) :|: s' >= 0, s' <= 1, z1 >= 0, z3 = 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 6 + i' + s1 }-> if2(s2, z', z'', z1, z2, 1 + x2 + y'' + i') :|: s1 >= 0, s1 <= i', s2 >= 0, s2 <= 1, z1 >= 0, z'' >= 0, z2 >= 0, y'' >= 0, z = 0, x2 >= 0, z3 = 1 + x2 + y'' + i', z' >= 0, i' >= 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: z3 >= 0, z1 >= 0, z = 1, z' >= 0, z'' >= 0, z2 >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s10, reach(v', z'', 1 + z1, z3, z3))), and(1, if1(s11, v, z'', 1 + z1, z3, z3))) :|: s10 >= 0, s10 <= 1, s11 >= 0, s11 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z' = 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 0, z'', z1, i'', z3), and(s12, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s13, v, z'', 1 + z1, z3, z3))) :|: s12 >= 0, s12 <= 1, s13 >= 0, s13 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z' = 0, z2 = 1 + (1 + x6) + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, x6 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + u' + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s14, reach(v', z'', 1 + z1, z3, z3))), and(0, if1(s15, v, z'', 1 + z1, z3, z3))) :|: s14 >= 0, s14 <= 1, s15 >= 0, s15 <= 1, v >= 0, z1 >= 0, z = 1, z'' >= 0, u' >= 0, z2 = 1 + 0 + v + (1 + u' + v' + i''), z3 >= 0, i'' >= 0, z' - 1 >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 13 + u' + y2 + z'' }-> or(or(if2(1, 1 + (z' - 1), z'', z1, i'', z3), and(s16, reach(v', z'', 1 + z1, z3, z3))), and(s17, if1(s18, v, z'', 1 + z1, z3, z3))) :|: s16 >= 0, s16 <= 1, s17 >= 0, s17 <= 1, s18 >= 0, s18 <= 1, v >= 0, z' - 1 >= 0, z1 >= 0, z = 1, z2 = 1 + (1 + y2) + v + (1 + u' + v' + i''), z'' >= 0, u' >= 0, y2 >= 0, z3 >= 0, i'' >= 0, v' >= 0 if2(z, z', z'', z1, z2, z3) -{ 10 + y1 + z'' }-> or(0, and(s8, if1(s9, v, z'', 1 + z1, z3, z3))) :|: s8 >= 0, s8 <= 1, s9 >= 0, s9 <= 1, y1 >= 0, z3 >= 0, z' - 1 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, z2 = 1 + (1 + y1) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(1, if1(s5, z2 - 1, z'', 1 + z1, z3, z3))) :|: s5 >= 0, s5 <= 1, z3 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0, z' = 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s6, v, z'', 1 + z1, z3, z3))) :|: s6 >= 0, s6 <= 1, z3 >= 0, v >= 0, z1 >= 0, z = 1, z'' >= 0, x3 >= 0, z' = 0, z2 = 1 + (1 + x3) + v + 0 if2(z, z', z'', z1, z2, z3) -{ 7 + z'' }-> or(0, and(0, if1(s7, z2 - 1, z'', 1 + z1, z3, z3))) :|: s7 >= 0, s7 <= 1, z3 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z1 >= 0, z = 1, z'' >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z' >= 0, z'' >= 0, z2 >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: z3 >= 0, z1 >= 0, z = 1, z2 = 0, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 or(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0 reach(z, z', z'', z1, z2) -{ 4 + z' }-> if1(s4, 1 + (z - 1), 1 + (z' - 1), z'', z1, z2) :|: s4 >= 0, s4 <= 1, z2 >= 0, z - 1 >= 0, z'' >= 0, z1 >= 0, z' - 1 >= 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(1, 0, 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z = 0, z' = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 0, 1 + (z' - 1), z'', z1, z2) :|: z2 >= 0, z'' >= 0, z' - 1 >= 0, z1 >= 0, z = 0 reach(z, z', z'', z1, z2) -{ 2 }-> if1(0, 1 + (z - 1), 0, z'', z1, z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 reachable(z, z', z'') -{ 1 }-> reach(z, z', 0, z'', z'') :|: z >= 0, z' >= 0, z'' >= 0 size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 2 + i }-> 1 + s'' :|: s'' >= 0, s'' <= i, x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Function symbols to be analyzed: {if2,reach,if1}, {reachable} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] size: runtime: O(n^1) [1 + z], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] and: runtime: O(1) [1], size: O(n^1) [z'] or: runtime: O(1) [1], size: O(n^1) [1 + z'] if2: runtime: INF, size: INF reach: runtime: ?, size: INF if1: runtime: ?, size: INF ---------------------------------------- (57) 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: none And the following fresh constants: none ---------------------------------------- (58) 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: eq(0, 0) -> true [1] eq(0, s(x)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] or(true, y) -> true [1] or(false, y) -> y [1] and(true, y) -> y [1] and(false, y) -> false [1] size(empty) -> 0 [1] size(edge(x, y, i)) -> s(size(i)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] reachable(x, y, i) -> reach(x, y, 0, i, i) [1] reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j) [1] if1(true, x, y, c, i, j) -> true [1] if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j) [1] if2(false, x, y, c, i, j) -> false [1] if2(true, x, y, c, empty, j) -> false [1] if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false or :: true:false -> true:false -> true:false and :: true:false -> true:false -> true:false size :: empty:edge -> 0:s empty :: empty:edge edge :: 0:s -> 0:s -> empty:edge -> empty:edge le :: 0:s -> 0:s -> true:false reachable :: 0:s -> 0:s -> empty:edge -> true:false reach :: 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if1 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false if2 :: true:false -> 0:s -> 0:s -> 0:s -> empty:edge -> empty:edge -> true:false Rewrite Strategy: INNERMOST ---------------------------------------- (59) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 empty => 0 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> y :|: z = 1, y >= 0, z' = y and(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> if2(le(c, size(j)), x, y, c, i, j) :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0 if1(z, z', z'', z1, z2, z3) -{ 1 }-> 1 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z3 = j, i >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> or(if2(1, x, y, c, i, j), and(eq(x, u), reach(v, y, 1 + c, j, j))) :|: v >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, y >= 0, z3 = j, i >= 0, u >= 0, j >= 0, z2 = 1 + u + v + i, x >= 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z2 = i, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0, z3 = j, i >= 0, z = 0 if2(z, z', z'', z1, z2, z3) -{ 1 }-> 0 :|: j >= 0, z' = x, z'' = y, z1 = c, c >= 0, z = 1, z2 = 0, x >= 0, y >= 0, z3 = j le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 or(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y or(z, z') -{ 1 }-> 1 :|: z = 1, y >= 0, z' = y reach(z, z', z'', z1, z2) -{ 1 }-> if1(eq(x, y), x, y, c, i, j) :|: j >= 0, c >= 0, z2 = j, x >= 0, y >= 0, i >= 0, z = x, z' = y, z'' = c, z1 = i reachable(z, z', z'') -{ 1 }-> reach(x, y, 0, i, i) :|: z'' = i, x >= 0, y >= 0, i >= 0, z = x, z' = y size(z) -{ 1 }-> 0 :|: z = 0 size(z) -{ 1 }-> 1 + size(i) :|: x >= 0, y >= 0, z = 1 + x + y + i, i >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (61) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) or(true, z0) -> true or(false, z0) -> z0 and(true, z0) -> z0 and(false, z0) -> false size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2) reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4) if1(true, z0, z1, z2, z3, z4) -> true if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4) if2(false, z0, z1, z2, z3, z4) -> false if2(true, z0, z1, z2, empty, z3) -> false if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))) Tuples: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) OR(true, z0) -> c4 OR(false, z0) -> c5 AND(true, z0) -> c6 AND(false, z0) -> c7 SIZE(empty) -> c8 SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(0, z0) -> c10 LE(s(z0), 0) -> c11 LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACHABLE(z0, z1, z2) -> c13(REACH(z0, z1, 0, z2, z2)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(true, z0, z1, z2, z3, z4) -> c15 IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(false, z0, z1, z2, z3, z4) -> c17 IF2(true, z0, z1, z2, empty, z3) -> c18 IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6)) S tuples: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) OR(true, z0) -> c4 OR(false, z0) -> c5 AND(true, z0) -> c6 AND(false, z0) -> c7 SIZE(empty) -> c8 SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(0, z0) -> c10 LE(s(z0), 0) -> c11 LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACHABLE(z0, z1, z2) -> c13(REACH(z0, z1, 0, z2, z2)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(true, z0, z1, z2, z3, z4) -> c15 IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(false, z0, z1, z2, z3, z4) -> c17 IF2(true, z0, z1, z2, empty, z3) -> c18 IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6)) K tuples:none Defined Rule Symbols: eq_2, or_2, and_2, size_1, le_2, reachable_3, reach_5, if1_6, if2_6 Defined Pair Symbols: EQ_2, OR_2, AND_2, SIZE_1, LE_2, REACHABLE_3, REACH_5, IF1_6, IF2_6 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6, c7, c8, c9_1, c10, c11, c12_1, c13_1, c14_2, c15, c16_3, c17, c18, c19_2, c20_3, c21_3 ---------------------------------------- (63) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: REACHABLE(z0, z1, z2) -> c13(REACH(z0, z1, 0, z2, z2)) Removed 13 trailing nodes: LE(0, z0) -> c10 IF1(true, z0, z1, z2, z3, z4) -> c15 SIZE(empty) -> c8 AND(false, z0) -> c7 OR(true, z0) -> c4 EQ(0, 0) -> c IF2(false, z0, z1, z2, z3, z4) -> c17 LE(s(z0), 0) -> c11 AND(true, z0) -> c6 OR(false, z0) -> c5 EQ(s(z0), 0) -> c2 EQ(0, s(z0)) -> c1 IF2(true, z0, z1, z2, empty, z3) -> c18 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) or(true, z0) -> true or(false, z0) -> z0 and(true, z0) -> z0 and(false, z0) -> false size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2) reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4) if1(true, z0, z1, z2, z3, z4) -> true if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4) if2(false, z0, z1, z2, z3, z4) -> false if2(true, z0, z1, z2, empty, z3) -> false if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(OR(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))), AND(eq(z0, z3), reach(z4, z1, s(z2), z6, z6)), REACH(z4, z1, s(z2), z6, z6)) K tuples:none Defined Rule Symbols: eq_2, or_2, and_2, size_1, le_2, reachable_3, reach_5, if1_6, if2_6 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6 Compound Symbols: c3_1, c9_1, c12_1, c14_2, c16_3, c19_2, c20_3, c21_3 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) or(true, z0) -> true or(false, z0) -> z0 and(true, z0) -> z0 and(false, z0) -> false size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2) reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4) if1(true, z0, z1, z2, z3, z4) -> true if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4) if2(false, z0, z1, z2, z3, z4) -> false if2(true, z0, z1, z2, empty, z3) -> false if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) K tuples:none Defined Rule Symbols: eq_2, or_2, and_2, size_1, le_2, reachable_3, reach_5, if1_6, if2_6 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6 Compound Symbols: c3_1, c9_1, c12_1, c14_2, c16_3, c19_1, c20_1, c21_1 ---------------------------------------- (67) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: or(true, z0) -> true or(false, z0) -> z0 and(true, z0) -> z0 and(false, z0) -> false reachable(z0, z1, z2) -> reach(z0, z1, 0, z2, z2) reach(z0, z1, z2, z3, z4) -> if1(eq(z0, z1), z0, z1, z2, z3, z4) if1(true, z0, z1, z2, z3, z4) -> true if1(false, z0, z1, z2, z3, z4) -> if2(le(z2, size(z4)), z0, z1, z2, z3, z4) if2(false, z0, z1, z2, z3, z4) -> false if2(true, z0, z1, z2, empty, z3) -> false if2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> or(if2(true, z0, z1, z2, z5, z6), and(eq(z0, z3), reach(z4, z1, s(z2), z6, z6))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) K tuples:none Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6 Compound Symbols: c3_1, c9_1, c12_1, c14_2, c16_3, c19_1, c20_1, c21_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) We considered the (Usable) Rules: eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_3 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = [1] + x_3 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 POL(SIZE(x_1)) = 0 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(empty) = [1] POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 POL(size(x_1)) = [1] POL(true) = [1] ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, REACH_5, IF1_6, IF2_6 Compound Symbols: c3_1, c9_1, c12_1, c14_2, c16_3, c19_1, c20_1, c21_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REACH(z0, z1, z2, z3, z4) -> c14(IF1(eq(z0, z1), z0, z1, z2, z3, z4), EQ(z0, z1)) by REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0))) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0))) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0))) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF1_6, IF2_6, REACH_5 Compound Symbols: c3_1, c9_1, c12_1, c16_3, c19_1, c20_1, c21_1, c14_2 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REACH(0, 0, x2, x3, x4) -> c14(IF1(true, 0, 0, x2, x3, x4), EQ(0, 0)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0))) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4), EQ(0, s(z0))) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4), EQ(s(z0), 0)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF1_6, IF2_6, REACH_5 Compound Symbols: c3_1, c9_1, c12_1, c16_3, c19_1, c20_1, c21_1, c14_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF1_6, IF2_6, REACH_5 Compound Symbols: c3_1, c9_1, c12_1, c16_3, c19_1, c20_1, c21_1, c14_2, c14_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF1(false, z0, z1, z2, z3, z4) -> c16(IF2(le(z2, size(z4)), z0, z1, z2, z3, z4), LE(z2, size(z4)), SIZE(z4)) by IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), LE(0, size(x4)), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)), SIZE(empty)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), LE(0, size(x4)), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)), SIZE(empty)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), LE(0, size(x4)), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty)), SIZE(empty)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_4 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = x_3 POL(SIZE(x_1)) = 0 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = x_1 + x_2 POL(empty) = 0 POL(eq(x_1, x_2)) = [1] + x_1 + x_2 POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = 0 POL(size(x_1)) = 0 POL(true) = [1] ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REACH(s(z0), s(z1), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(z0), s(z1), x2, x3, x4), EQ(s(z0), s(z1))) by REACH(s(0), s(0), x2, x3, x4) -> c14(IF1(true, s(0), s(0), x2, x3, x4), EQ(s(0), s(0))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(0), x2, x3, x4) -> c14(IF1(true, s(0), s(0), x2, x3, x4), EQ(s(0), s(0))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(0), x2, x3, x4) -> c14(IF1(true, s(0), s(0), x2, x3, x4), EQ(s(0), s(0))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c16_2, c14_2 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c16_2, c14_2 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) We considered the (Usable) Rules: eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_4 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = [1] + x_2 + x_4 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 POL(SIZE(x_1)) = 0 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(empty) = 0 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = 0 POL(size(x_1)) = 0 POL(true) = [1] ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c16_2, c14_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF1(false, x0, x1, x2, x3, edge(z0, z1, z2)) -> c16(IF2(le(x2, s(size(z2))), x0, x1, x2, x3, edge(z0, z1, z2)), LE(x2, size(edge(z0, z1, z2))), SIZE(edge(z0, z1, z2))) by IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), LE(0, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), LE(0, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), LE(0, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c16(LE(x2, size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3 ---------------------------------------- (93) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1 ---------------------------------------- (95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_4 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [2]x_3 POL(SIZE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = [2] POL(empty) = 0 POL(eq(x_1, x_2)) = [3] + [3]x_1 + [3]x_2 POL(false) = 0 POL(le(x_1, x_2)) = [3] POL(s(x_1)) = 0 POL(size(x_1)) = [2] + [2]x_1 POL(true) = 0 ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1 ---------------------------------------- (97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) We considered the (Usable) Rules: eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = [1] + x_3 POL(SIZE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = x_2 POL(empty) = 0 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(size(x_1)) = 0 POL(true) = [1] ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF1(false, x0, x1, x2, x3, empty) -> c16(IF2(le(x2, 0), x0, x1, x2, x3, empty), LE(x2, size(empty))) by IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty), LE(0, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(IF2(false, x0, x1, s(z0), x3, empty), LE(s(z0), size(empty))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty), LE(0, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(IF2(false, x0, x1, s(z0), x3, empty), LE(s(z0), size(empty))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty), LE(0, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(IF2(false, x0, x1, s(z0), x3, empty), LE(s(z0), size(empty))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1 ---------------------------------------- (103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_2 + x_6 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = x_2 + x_6 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5 POL(SIZE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = x_1 + x_2 POL(empty) = [1] POL(eq(x_1, x_2)) = [1] + x_1 + x_2 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(size(x_1)) = 0 POL(true) = [1] ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) We considered the (Usable) Rules:none And the Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6)) = x_4 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(LE(x_1, x_2)) = 0 POL(REACH(x_1, x_2, x_3, x_4, x_5)) = x_3 POL(SIZE(x_1)) = 0 POL(c(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c19(x_1)) = x_1 POL(c20(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(edge(x_1, x_2, x_3)) = x_1 + x_2 POL(empty) = 0 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(size(x_1)) = 0 POL(true) = [1] ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1 ---------------------------------------- (107) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(0, s(z0), x2, x3, x4) -> c14(IF1(false, 0, s(z0), x2, x3, x4)) by REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, x0, x1, s(z0), x3, edge(x4, x5, x6)) -> c16(IF2(le(z0, size(x6)), x0, x1, s(z0), x3, edge(x4, x5, x6)), LE(s(z0), size(edge(x4, x5, x6))), SIZE(edge(x4, x5, x6))) by IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c14_2, c16_3, c_1, c16_1 ---------------------------------------- (111) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(s(z0), 0, x2, x3, x4) -> c14(IF1(false, s(z0), 0, x2, x3, x4)) by REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, IF1_6, REACH_5 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c16_2, c14_2, c14_1, c16_3, c_1, c16_1 ---------------------------------------- (113) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, x0, x1, 0, x3, x4) -> c16(IF2(true, x0, x1, 0, x3, x4), SIZE(x4)) by IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2, c_1, c16_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, empty)) -> c16(IF2(le(x2, s(0)), x0, x1, x2, x3, edge(x4, x5, empty)), LE(x2, size(edge(x4, x5, empty))), SIZE(edge(x4, x5, empty))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_3, c16_2, c_1, c16_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))) -> c16(IF2(le(x2, s(s(size(z2)))), x0, x1, x2, x3, edge(x4, x5, edge(z0, z1, z2))), LE(x2, size(edge(x4, x5, edge(z0, z1, z2)))), SIZE(edge(x4, x5, edge(z0, z1, z2)))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (119) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(s(0), s(s(z0)), x2, x3, x4) -> c14(IF1(false, s(0), s(s(z0)), x2, x3, x4), EQ(s(0), s(s(z0)))) by REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (121) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF1(false, s(0), s(s(x0)), 0, x2, x3) -> c16(IF2(true, s(0), s(s(x0)), 0, x2, x3), SIZE(x3)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) by IF1(false, z0, z1, z2, z3, empty) -> c16(LE(z2, 0)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, empty) -> c16(LE(z2, 0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(LE(x2, size(edge(x4, x5, x6)))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, empty) -> c16(LE(x2, size(empty))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF1(false, z0, z1, z2, z3, empty) -> c16(LE(z2, 0)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) by IF1(false, z0, z1, s(z2), z3, empty) -> c16(LE(s(z2), 0)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, s(z2), z3, empty) -> c16(LE(s(z2), 0)) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, s(z0), x3, empty) -> c16(LE(s(z0), size(empty))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF1(false, z0, z1, s(z2), z3, empty) -> c16(LE(s(z2), 0)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(size(empty))), SIZE(edge(z4, z5, empty))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(size(edge(z6, z7, z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) by IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (137) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(s(s(z0)), s(0), x2, x3, x4) -> c14(IF1(false, s(s(z0)), s(0), x2, x3, x4), EQ(s(s(z0)), s(0))) by REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (139) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF1(false, s(s(x0)), s(0), 0, x2, x3) -> c16(IF2(true, s(s(x0)), s(0), 0, x2, x3), SIZE(x3)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_2, c14_1, c16_2, c_1, c16_1, c16_3 ---------------------------------------- (141) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(s(s(z0)), s(s(z1)), x2, x3, x4) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), x2, x3, x4), EQ(s(s(z0)), s(s(z1)))) by REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_2, c_1, c16_1, c16_3, c14_2 ---------------------------------------- (143) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: IF1(false, x0, x1, 0, x3, edge(x4, x5, x6)) -> c16(IF2(true, x0, x1, 0, x3, edge(x4, x5, x6)), SIZE(edge(x4, x5, x6))) IF1(false, x0, x1, 0, x3, empty) -> c16(IF2(true, x0, x1, 0, x3, empty)) IF1(false, s(s(x0)), s(s(x1)), 0, x3, x4) -> c16(IF2(true, s(s(x0)), s(s(x1)), 0, x3, x4), SIZE(x4)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c_1, c16_3, c14_2 ---------------------------------------- (145) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(s(x0), s(x1), x2, x3, x4) -> c14(EQ(s(x0), s(x1))) by REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c_1, c16_3, c14_2 ---------------------------------------- (147) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace REACH(s(0), s(0), x2, x3, x4) -> c14(EQ(s(0), s(0))) by REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, IF1_6, REACH_5 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c_1, c14_1, c16_3, c14_2 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, x0, x1, x2, x3, edge(x4, x5, x6)) -> c(SIZE(edge(x4, x5, x6))) by IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c14_2, c_1 ---------------------------------------- (151) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, s(z2), z3, edge(z4, z5, z6)) -> c16(IF2(le(z2, size(z6)), z0, z1, s(z2), z3, edge(z4, z5, z6)), LE(s(z2), s(size(z6))), SIZE(edge(z4, z5, z6))) by IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) S tuples: EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: EQ_2, SIZE_1, LE_2, IF2_6, REACH_5, IF1_6 Compound Symbols: c3_1, c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) S tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2 Compound Symbols: c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1 ---------------------------------------- (155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: REACH(s(0), s(0), s(x2), x6, x6) -> c14(EQ(s(0), s(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) S tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6), EQ(s(0), s(s(z0)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6), EQ(s(s(z0)), s(0))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2 Compound Symbols: c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) S tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2 Compound Symbols: c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c_1, c16_3, c14_2, c3_1 ---------------------------------------- (159) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, z6)) -> c(LE(z2, s(size(z6)))) by IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) S tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2 Compound Symbols: c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c14_2, c_1, c3_1 ---------------------------------------- (161) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, empty)) -> c16(IF2(le(z2, s(0)), z0, z1, z2, z3, edge(z4, z5, empty)), LE(z2, s(0)), SIZE(edge(z4, z5, empty))) by IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) S tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2 Compound Symbols: c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c16_3, c14_2, c_1, c3_1 ---------------------------------------- (163) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(false, z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(z2, s(s(size(z8)))), z0, z1, z2, z3, edge(z4, z5, edge(z6, z7, z8))), LE(z2, s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) by IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) S tuples: SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: SIZE_1, LE_2, IF2_6, REACH_5, IF1_6, EQ_2 Compound Symbols: c9_1, c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SIZE(edge(z0, z1, z2)) -> c9(SIZE(z2)) by SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) S tuples: LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0)), SIZE(edge(z4, z5, empty))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: LE_2, IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1 Compound Symbols: c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) S tuples: LE(s(z0), s(z1)) -> c12(LE(z0, z1)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: LE_2, IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1 Compound Symbols: c12_1, c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c16_2 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c12(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) S tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x2), s(0))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)), LE(s(x1), s(0))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2 Compound Symbols: c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c16_2, c12_1 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) S tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2 Compound Symbols: c19_1, c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c19(IF2(true, z0, z1, z2, z5, z6)) by IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) S tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) K tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2 Compound Symbols: c20_1, c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c20(EQ(z0, z3)) by IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) S tuples: IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) K tuples: REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF2_6, REACH_5, IF1_6, EQ_2, SIZE_1, LE_2 Compound Symbols: c21_1, c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1 ---------------------------------------- (177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF2(true, z0, z1, z2, edge(z3, z4, z5), z6) -> c21(REACH(z4, z1, s(z2), z6, z6)) by IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) S tuples: REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) K tuples: REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REACH(0, s(z0), s(x2), x6, x6) -> c14(IF1(false, 0, s(z0), s(x2), x6, x6)) by REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) S tuples: REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_1, c14_2, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REACH(s(z0), 0, s(x2), x6, x6) -> c14(IF1(false, s(z0), 0, s(x2), x6, x6)) by REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c14_1, c_1, c16_3, c3_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REACH(s(z0), s(z1), s(x2), x6, x6) -> c14(EQ(s(z0), s(z1))) by REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c_1, c16_3, c3_1, c14_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(SIZE(edge(z4, z5, z6))) by IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, EQ_2, SIZE_1, LE_2, IF2_6 Compound Symbols: c14_2, c16_3, c3_1, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1 ---------------------------------------- (195) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c3(EQ(s(y0), s(y1))) by EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) S tuples: REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) K tuples: IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: REACH_5, IF1_6, SIZE_1, LE_2, IF2_6, EQ_2 Compound Symbols: c14_2, c16_3, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c3_1 ---------------------------------------- (197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REACH(s(s(z0)), s(s(z1)), s(x2), x6, x6) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(x2), x6, x6), EQ(s(s(z0)), s(s(z1)))) by REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) S tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) K tuples: IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF1_6, REACH_5, SIZE_1, LE_2, IF2_6, EQ_2 Compound Symbols: c16_3, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c3_1, c14_2 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REACH(s(0), s(s(z0)), s(x2), x6, x6) -> c14(IF1(false, s(0), s(s(z0)), s(x2), x6, x6)) by REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) S tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF1_6, REACH_5, SIZE_1, LE_2, IF2_6, EQ_2 Compound Symbols: c16_3, c14_1, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c3_1, c14_2 ---------------------------------------- (201) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REACH(s(s(z0)), s(0), s(x2), x6, x6) -> c14(IF1(false, s(s(z0)), s(0), s(x2), x6, x6)) by REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) size(empty) -> 0 size(edge(z0, z1, z2)) -> s(size(z2)) Tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x2), s(size(z6)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c(LE(s(x1), s(size(z6)))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) S tuples: IF1(false, 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), 0, s(x0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(x0), 0, s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(0), s(s(x0)), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x1, size(z6)), s(s(x0)), s(0), s(x1), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x1), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)) -> c16(IF2(le(x2, size(z6)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, z6), edge(z4, z5, z6)), LE(s(x2), s(size(z6))), SIZE(edge(z4, z5, z6))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), 0, s(x0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(x0), 0, s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x2), s(s(size(z8)))), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x2), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(0), s(s(x0)), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))) -> c16(IF2(le(s(x1), s(s(size(z8)))), s(s(x0)), s(0), s(x1), edge(z4, z5, edge(z6, z7, z8)), edge(z4, z5, edge(z6, z7, z8))), LE(s(x1), s(s(size(z8)))), SIZE(edge(z4, z5, edge(z6, z7, z8)))) SIZE(edge(z0, z1, edge(y0, y1, y2))) -> c9(SIZE(edge(y0, y1, y2))) LE(s(s(y0)), s(s(y1))) -> c12(LE(s(y0), s(y1))) IF1(false, 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), 0, s(x0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(x0), 0, s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x2), s(0)), s(s(x0)), s(s(x1)), s(x2), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(0), s(s(x0)), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF1(false, s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty)) -> c16(IF2(le(s(x1), s(0)), s(s(x0)), s(0), s(x1), edge(z4, z5, empty), edge(z4, z5, empty))) IF2(true, z0, z1, z2, edge(z3, z4, edge(y3, y4, y5)), z6) -> c19(IF2(true, z0, z1, z2, edge(y3, y4, y5), z6)) IF2(true, z0, s(y0), z2, edge(z3, 0, z5), z6) -> c21(REACH(0, s(y0), s(z2), z6, z6)) IF2(true, z0, 0, z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), 0, s(z2), z6, z6)) IF2(true, z0, s(s(y1)), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(s(y1)), s(z2), z6, z6)) IF2(true, z0, s(y1), z2, edge(z3, s(y0), z5), z6) -> c21(REACH(s(y0), s(y1), s(z2), z6, z6)) IF2(true, z0, s(s(y0)), z2, edge(z3, s(0), z5), z6) -> c21(REACH(s(0), s(s(y0)), s(z2), z6, z6)) IF2(true, z0, s(0), z2, edge(z3, s(s(y0)), z5), z6) -> c21(REACH(s(s(y0)), s(0), s(z2), z6, z6)) REACH(0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, 0, s(z0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(z0), 0, s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c3(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, y6), edge(y4, y5, y6)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, edge(y6, y7, y8)), edge(y4, y5, edge(y6, y7, y8))), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)) -> c14(IF1(eq(z0, z1), s(s(z0)), s(s(z1)), s(z2), edge(y4, y5, empty), edge(y4, y5, empty)), EQ(s(s(z0)), s(s(z1)))) REACH(s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3) -> c14(IF1(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), s(z2), z3, z3), EQ(s(s(s(y0))), s(s(s(y1))))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(0), s(s(z0)), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, y4), edge(y2, y3, y4))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6))) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, edge(y4, y5, y6)), edge(y2, y3, edge(y4, y5, y6)))) REACH(s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty)) -> c14(IF1(false, s(s(z0)), s(0), s(z1), edge(y2, y3, empty), edge(y2, y3, empty))) K tuples: IF2(true, s(s(y0)), z1, z2, edge(s(s(y1)), z4, z5), z6) -> c20(EQ(s(s(y0)), s(s(y1)))) REACH(s(s(y0)), s(s(y1)), s(z2), z3, z3) -> c14(EQ(s(s(y0)), s(s(y1)))) IF1(false, 0, s(z0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(z0), 0, s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(0), s(s(z0)), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(0), s(z1), edge(z2, z3, edge(y2, y3, y4)), edge(z2, z3, edge(y2, y3, y4))) -> c(SIZE(edge(z2, z3, edge(y2, y3, y4)))) IF1(false, s(s(z0)), s(s(z1)), s(z2), edge(z3, z4, edge(y2, y3, y4)), edge(z3, z4, edge(y2, y3, y4))) -> c(SIZE(edge(z3, z4, edge(y2, y3, y4)))) Defined Rule Symbols: eq_2, le_2, size_1 Defined Pair Symbols: IF1_6, SIZE_1, LE_2, IF2_6, REACH_5, EQ_2 Compound Symbols: c16_3, c_1, c9_1, c12_1, c16_1, c19_1, c20_1, c21_1, c14_1, c3_1, c14_2