927.52/292.39 WORST_CASE(Omega(n^1), O(n^2)) 927.57/292.41 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 927.57/292.41 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 927.57/292.41 927.57/292.41 927.57/292.41 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 927.57/292.41 927.57/292.41 (0) CpxRelTRS 927.57/292.41 (1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 178 ms] 927.57/292.41 (2) CpxRelTRS 927.57/292.41 (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] 927.57/292.41 (4) CdtProblem 927.57/292.41 (5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] 927.57/292.41 (6) CdtProblem 927.57/292.41 (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (8) CdtProblem 927.57/292.41 (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 555 ms] 927.57/292.41 (10) CdtProblem 927.57/292.41 (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 299 ms] 927.57/292.41 (12) CdtProblem 927.57/292.41 (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 272 ms] 927.57/292.41 (14) CdtProblem 927.57/292.41 (15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (16) CdtProblem 927.57/292.41 (17) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (18) CdtProblem 927.57/292.41 (19) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (20) CdtProblem 927.57/292.41 (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 623 ms] 927.57/292.41 (22) CdtProblem 927.57/292.41 (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 756 ms] 927.57/292.41 (24) CdtProblem 927.57/292.41 (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (26) CdtProblem 927.57/292.41 (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (28) CdtProblem 927.57/292.41 (29) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] 927.57/292.41 (30) CdtProblem 927.57/292.41 (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (32) CdtProblem 927.57/292.41 (33) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1710 ms] 927.57/292.41 (34) CdtProblem 927.57/292.41 (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2376 ms] 927.57/292.41 (36) CdtProblem 927.57/292.41 (37) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 3135 ms] 927.57/292.41 (38) CdtProblem 927.57/292.41 (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (40) CdtProblem 927.57/292.41 (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (42) CdtProblem 927.57/292.41 (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] 927.57/292.41 (44) CdtProblem 927.57/292.41 (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (46) CdtProblem 927.57/292.41 (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (48) CdtProblem 927.57/292.41 (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (50) CdtProblem 927.57/292.41 (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (52) CdtProblem 927.57/292.41 (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (54) CdtProblem 927.57/292.41 (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (56) CdtProblem 927.57/292.41 (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (58) CdtProblem 927.57/292.41 (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (60) CdtProblem 927.57/292.41 (61) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (62) CdtProblem 927.57/292.41 (63) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (64) CdtProblem 927.57/292.41 (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (66) CdtProblem 927.57/292.41 (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (68) CdtProblem 927.57/292.41 (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (70) CdtProblem 927.57/292.41 (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (72) CdtProblem 927.57/292.41 (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (74) CdtProblem 927.57/292.41 (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (76) CdtProblem 927.57/292.41 (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (78) CdtProblem 927.57/292.41 (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (80) CdtProblem 927.57/292.41 (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (82) CdtProblem 927.57/292.41 (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (84) CdtProblem 927.57/292.41 (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (86) CdtProblem 927.57/292.41 (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (88) CdtProblem 927.57/292.41 (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (90) CdtProblem 927.57/292.41 (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (92) CdtProblem 927.57/292.41 (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (94) CdtProblem 927.57/292.41 (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (96) CdtProblem 927.57/292.41 (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (98) CdtProblem 927.57/292.41 (99) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] 927.57/292.41 (100) CdtProblem 927.57/292.41 (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (102) CdtProblem 927.57/292.41 (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (104) CdtProblem 927.57/292.41 (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (106) CdtProblem 927.57/292.41 (107) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (108) CdtProblem 927.57/292.41 (109) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (110) CdtProblem 927.57/292.41 (111) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (112) CdtProblem 927.57/292.41 (113) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (114) CdtProblem 927.57/292.41 (115) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] 927.57/292.41 (116) CdtProblem 927.57/292.41 (117) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (118) CdtProblem 927.57/292.41 (119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (120) CdtProblem 927.57/292.41 (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 5240 ms] 927.57/292.41 (122) CdtProblem 927.57/292.41 (123) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] 927.57/292.41 (124) BOUNDS(1, 1) 927.57/292.41 (125) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 927.57/292.41 (126) TRS for Loop Detection 927.57/292.41 (127) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 927.57/292.41 (128) BEST 927.57/292.41 (129) proven lower bound 927.57/292.41 (130) LowerBoundPropagationProof [FINISHED, 0 ms] 927.57/292.41 (131) BOUNDS(n^1, INF) 927.57/292.41 (132) TRS for Loop Detection 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (0) 927.57/292.41 Obligation: 927.57/292.41 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 927.57/292.41 927.57/292.41 927.57/292.41 The TRS R consists of the following rules: 927.57/292.41 927.57/292.41 mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) 927.57/292.41 mergesort(Cons(x, Nil)) -> Cons(x, Nil) 927.57/292.41 merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) 927.57/292.41 merge(Cons(x, xs), Nil) -> Cons(x, xs) 927.57/292.41 splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) 927.57/292.41 splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 merge(Nil, xs2) -> xs2 927.57/292.41 notEmpty(Cons(x, xs)) -> True 927.57/292.41 notEmpty(Nil) -> False 927.57/292.41 goal(xs) -> mergesort(xs) 927.57/292.41 927.57/292.41 The (relative) TRS S consists of the following rules: 927.57/292.41 927.57/292.41 <=(S(x), S(y)) -> <=(x, y) 927.57/292.41 <=(0, y) -> True 927.57/292.41 <=(S(x), 0) -> False 927.57/292.41 merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) 927.57/292.41 merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) 927.57/292.41 927.57/292.41 Rewrite Strategy: INNERMOST 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) 927.57/292.41 proved termination of relative rules 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (2) 927.57/292.41 Obligation: 927.57/292.41 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 927.57/292.41 927.57/292.41 927.57/292.41 The TRS R consists of the following rules: 927.57/292.41 927.57/292.41 mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) 927.57/292.41 mergesort(Cons(x, Nil)) -> Cons(x, Nil) 927.57/292.41 merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) 927.57/292.41 merge(Cons(x, xs), Nil) -> Cons(x, xs) 927.57/292.41 splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) 927.57/292.41 splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 merge(Nil, xs2) -> xs2 927.57/292.41 notEmpty(Cons(x, xs)) -> True 927.57/292.41 notEmpty(Nil) -> False 927.57/292.41 goal(xs) -> mergesort(xs) 927.57/292.41 927.57/292.41 The (relative) TRS S consists of the following rules: 927.57/292.41 927.57/292.41 <=(S(x), S(y)) -> <=(x, y) 927.57/292.41 <=(0, y) -> True 927.57/292.41 <=(S(x), 0) -> False 927.57/292.41 merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) 927.57/292.41 merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) 927.57/292.41 927.57/292.41 Rewrite Strategy: INNERMOST 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (3) CpxTrsToCdtProof (UPPER BOUND(ID)) 927.57/292.41 Converted Cpx (relative) TRS to CDT 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (4) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 notEmpty(Cons(z0, z1)) -> True 927.57/292.41 notEmpty(Nil) -> False 927.57/292.41 goal(z0) -> mergesort(z0) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 <='(0, z0) -> c1 927.57/292.41 <='(S(z0), 0) -> c2 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGESORT(Cons(z0, Nil)) -> c6 927.57/292.41 MERGESORT(Nil) -> c7 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 NOTEMPTY(Cons(z0, z1)) -> c13 927.57/292.41 NOTEMPTY(Nil) -> c14 927.57/292.41 GOAL(z0) -> c15(MERGESORT(z0)) 927.57/292.41 S tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGESORT(Cons(z0, Nil)) -> c6 927.57/292.41 MERGESORT(Nil) -> c7 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 NOTEMPTY(Cons(z0, z1)) -> c13 927.57/292.41 NOTEMPTY(Nil) -> c14 927.57/292.41 GOAL(z0) -> c15(MERGESORT(z0)) 927.57/292.41 K tuples:none 927.57/292.41 Defined Rule Symbols: mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3, NOTEMPTY_1, GOAL_1 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c1, c2, c3_1, c4_1, c5_1, c6, c7, c8_2, c9, c10, c11_1, c12_3, c13, c14, c15_1 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (5) CdtLeafRemovalProof (ComplexityIfPolyImplication) 927.57/292.41 Removed 1 leading nodes: 927.57/292.41 GOAL(z0) -> c15(MERGESORT(z0)) 927.57/292.41 Removed 6 trailing nodes: 927.57/292.41 NOTEMPTY(Cons(z0, z1)) -> c13 927.57/292.41 NOTEMPTY(Nil) -> c14 927.57/292.41 MERGESORT(Cons(z0, Nil)) -> c6 927.57/292.41 MERGESORT(Nil) -> c7 927.57/292.41 <='(S(z0), 0) -> c2 927.57/292.41 <='(0, z0) -> c1 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (6) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 notEmpty(Cons(z0, z1)) -> True 927.57/292.41 notEmpty(Nil) -> False 927.57/292.41 goal(z0) -> mergesort(z0) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 S tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 K tuples:none 927.57/292.41 Defined Rule Symbols: mergesort_1, merge_2, splitmerge_3, notEmpty_1, goal_1, <=_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_3 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 The following rules are not usable and were removed: 927.57/292.41 notEmpty(Cons(z0, z1)) -> True 927.57/292.41 notEmpty(Nil) -> False 927.57/292.41 goal(z0) -> mergesort(z0) 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (8) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 S tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 K tuples:none 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_3 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.41 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 We considered the (Usable) Rules:none 927.57/292.41 And the Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 The order we found is given by the following interpretation: 927.57/292.41 927.57/292.41 Polynomial interpretation : 927.57/292.41 927.57/292.41 POL(0) = 0 927.57/292.41 POL(<=(x_1, x_2)) = 0 927.57/292.41 POL(<='(x_1, x_2)) = 0 927.57/292.41 POL(Cons(x_1, x_2)) = [1] + x_2 927.57/292.41 POL(False) = 0 927.57/292.41 POL(MERGE(x_1, x_2)) = 0 927.57/292.41 POL(MERGESORT(x_1)) = [2]x_1^2 927.57/292.41 POL(MERGE[ITE](x_1, x_2, x_3)) = 0 927.57/292.41 POL(Nil) = 0 927.57/292.41 POL(S(x_1)) = 0 927.57/292.41 POL(SPLITMERGE(x_1, x_2, x_3)) = [1] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 927.57/292.41 POL(True) = 0 927.57/292.41 POL(c(x_1)) = x_1 927.57/292.41 POL(c10) = 0 927.57/292.41 POL(c11(x_1)) = x_1 927.57/292.41 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 POL(c3(x_1)) = x_1 927.57/292.41 POL(c4(x_1)) = x_1 927.57/292.41 POL(c5(x_1)) = x_1 927.57/292.41 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(c9) = 0 927.57/292.41 POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 927.57/292.41 POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2 927.57/292.41 POL(mergesort(x_1)) = 0 927.57/292.41 POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (10) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_3 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.41 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 We considered the (Usable) Rules: 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 And the Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 The order we found is given by the following interpretation: 927.57/292.41 927.57/292.41 Polynomial interpretation : 927.57/292.41 927.57/292.41 POL(0) = 0 927.57/292.41 POL(<=(x_1, x_2)) = [1] 927.57/292.41 POL(<='(x_1, x_2)) = 0 927.57/292.41 POL(Cons(x_1, x_2)) = [1] + x_2 927.57/292.41 POL(False) = [1] 927.57/292.41 POL(MERGE(x_1, x_2)) = [1] 927.57/292.41 POL(MERGESORT(x_1)) = [2]x_1^2 927.57/292.41 POL(MERGE[ITE](x_1, x_2, x_3)) = x_1^2 927.57/292.41 POL(Nil) = 0 927.57/292.41 POL(S(x_1)) = 0 927.57/292.41 POL(SPLITMERGE(x_1, x_2, x_3)) = [2] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 927.57/292.41 POL(True) = [1] 927.57/292.41 POL(c(x_1)) = x_1 927.57/292.41 POL(c10) = 0 927.57/292.41 POL(c11(x_1)) = x_1 927.57/292.41 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 POL(c3(x_1)) = x_1 927.57/292.41 POL(c4(x_1)) = x_1 927.57/292.41 POL(c5(x_1)) = x_1 927.57/292.41 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(c9) = 0 927.57/292.41 POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 927.57/292.41 POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2 927.57/292.41 POL(mergesort(x_1)) = 0 927.57/292.41 POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (12) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_3 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.41 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 We considered the (Usable) Rules:none 927.57/292.41 And the Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 The order we found is given by the following interpretation: 927.57/292.41 927.57/292.41 Polynomial interpretation : 927.57/292.41 927.57/292.41 POL(0) = 0 927.57/292.41 POL(<=(x_1, x_2)) = 0 927.57/292.41 POL(<='(x_1, x_2)) = 0 927.57/292.41 POL(Cons(x_1, x_2)) = [1] + x_2 927.57/292.41 POL(False) = 0 927.57/292.41 POL(MERGE(x_1, x_2)) = 0 927.57/292.41 POL(MERGESORT(x_1)) = [2]x_1^2 927.57/292.41 POL(MERGE[ITE](x_1, x_2, x_3)) = 0 927.57/292.41 POL(Nil) = 0 927.57/292.41 POL(S(x_1)) = 0 927.57/292.41 POL(SPLITMERGE(x_1, x_2, x_3)) = [2]x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 927.57/292.41 POL(True) = 0 927.57/292.41 POL(c(x_1)) = x_1 927.57/292.41 POL(c10) = 0 927.57/292.41 POL(c11(x_1)) = x_1 927.57/292.41 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 POL(c3(x_1)) = x_1 927.57/292.41 POL(c4(x_1)) = x_1 927.57/292.41 POL(c5(x_1)) = x_1 927.57/292.41 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(c9) = 0 927.57/292.41 POL(merge(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 927.57/292.41 POL(merge[Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + x_2^2 927.57/292.41 POL(mergesort(x_1)) = 0 927.57/292.41 POL(splitmerge(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 + x_3^2 + x_2*x_3 + x_1*x_3 + x_1^2 + x_1*x_2 + x_2^2 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (14) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c8_2, c9, c10, c11_1, c12_3 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (15) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 Use narrowing to replace MERGE(Cons(z0, z1), Cons(z2, z3)) -> c8(MERGE[ITE](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)), <='(z0, z2)) by 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (16) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, MERGE_2, SPLITMERGE_3 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c9, c10, c11_1, c12_3, c8_2 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (17) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 Removed 2 trailing nodes: 927.57/292.41 MERGE(Nil, z0) -> c10 927.57/292.41 MERGE(Cons(z0, z1), Nil) -> c9 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (18) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3)), <='(0, z0)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3)), <='(S(z0), 0)) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_3, c8_2 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 Removed 2 trailing tuple parts 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (20) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_3, c8_2, c8_1 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.41 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 We considered the (Usable) Rules: 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 And the Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 The order we found is given by the following interpretation: 927.57/292.41 927.57/292.41 Polynomial interpretation : 927.57/292.41 927.57/292.41 POL(0) = 0 927.57/292.41 POL(<=(x_1, x_2)) = [1] 927.57/292.41 POL(<='(x_1, x_2)) = 0 927.57/292.41 POL(Cons(x_1, x_2)) = [2] + x_2 927.57/292.41 POL(False) = [1] 927.57/292.41 POL(MERGE(x_1, x_2)) = x_2 + x_1*x_2 927.57/292.41 POL(MERGESORT(x_1)) = x_1^2 927.57/292.41 POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 + x_1*x_3 927.57/292.41 POL(Nil) = 0 927.57/292.41 POL(S(x_1)) = 0 927.57/292.41 POL(SPLITMERGE(x_1, x_2, x_3)) = x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 927.57/292.41 POL(True) = 0 927.57/292.41 POL(c(x_1)) = x_1 927.57/292.41 POL(c11(x_1)) = x_1 927.57/292.41 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 POL(c3(x_1)) = x_1 927.57/292.41 POL(c4(x_1)) = x_1 927.57/292.41 POL(c5(x_1)) = x_1 927.57/292.41 POL(c8(x_1)) = x_1 927.57/292.41 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(merge(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 927.57/292.41 POL(mergesort(x_1)) = x_1 927.57/292.41 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (22) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_3, c8_2, c8_1 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.41 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 We considered the (Usable) Rules: 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 And the Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 The order we found is given by the following interpretation: 927.57/292.41 927.57/292.41 Polynomial interpretation : 927.57/292.41 927.57/292.41 POL(0) = 0 927.57/292.41 POL(<=(x_1, x_2)) = [1] 927.57/292.41 POL(<='(x_1, x_2)) = 0 927.57/292.41 POL(Cons(x_1, x_2)) = [1] + x_2 927.57/292.41 POL(False) = 0 927.57/292.41 POL(MERGE(x_1, x_2)) = [1] + x_2 927.57/292.41 POL(MERGESORT(x_1)) = [2]x_1^2 927.57/292.41 POL(MERGE[ITE](x_1, x_2, x_3)) = x_1 + x_3 927.57/292.41 POL(Nil) = 0 927.57/292.41 POL(S(x_1)) = 0 927.57/292.41 POL(SPLITMERGE(x_1, x_2, x_3)) = [1] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 927.57/292.41 POL(True) = [1] 927.57/292.41 POL(c(x_1)) = x_1 927.57/292.41 POL(c11(x_1)) = x_1 927.57/292.41 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 POL(c3(x_1)) = x_1 927.57/292.41 POL(c4(x_1)) = x_1 927.57/292.41 POL(c5(x_1)) = x_1 927.57/292.41 POL(c8(x_1)) = x_1 927.57/292.41 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(merge(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 927.57/292.41 POL(mergesort(x_1)) = [2]x_1 927.57/292.41 POL(splitmerge(x_1, x_2, x_3)) = [2]x_1 + [2]x_2 + [2]x_3 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (24) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c12_3, c8_2, c8_1 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 Use narrowing to replace SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) by 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Nil))) 927.57/292.41 SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0), MERGESORT(Nil)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil)), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil), MERGESORT(x1)) 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (26) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Nil))) 927.57/292.41 SPLITMERGE(Nil, x0, Nil) -> c12(MERGE(mergesort(x0), Nil), MERGESORT(x0), MERGESORT(Nil)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(Cons(z0, Nil)), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, Nil, x1) -> c12(MERGE(Nil, mergesort(x1)), MERGESORT(Nil), MERGESORT(x1)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_3 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 Removed 6 trailing tuple parts 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (28) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, Nil, x1) -> c12(MERGESORT(x1)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Nil, z0, z1) -> c12(MERGE(mergesort(z0), mergesort(z1)), MERGESORT(z0), MERGESORT(z1)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_3, c12_2, c12_1 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (29) CdtLeafRemovalProof (ComplexityIfPolyImplication) 927.57/292.41 Removed 1 leading nodes: 927.57/292.41 SPLITMERGE(Nil, Nil, x1) -> c12(MERGESORT(x1)) 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (30) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_2, c8_1, c12_3, c12_2, c12_1 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.41 Use narrowing to replace MERGE(Cons(S(z0), x1), Cons(S(z1), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(z0), x1), Cons(S(z1), x3)), <='(S(z0), S(z1))) by 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.41 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.41 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (32) 927.57/292.41 Obligation: 927.57/292.41 Complexity Dependency Tuples Problem 927.57/292.41 927.57/292.41 Rules: 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.41 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.41 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.41 S tuples: 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.41 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.41 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.41 K tuples: 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.41 927.57/292.41 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.41 927.57/292.41 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.41 927.57/292.41 927.57/292.41 ---------------------------------------- 927.57/292.41 927.57/292.41 (33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.41 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.41 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.41 We considered the (Usable) Rules: 927.57/292.41 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.41 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.41 <=(S(z0), 0) -> False 927.57/292.41 merge(Nil, z0) -> z0 927.57/292.41 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.41 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.41 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.41 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.41 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.41 mergesort(Nil) -> Nil 927.57/292.41 <=(0, z0) -> True 927.57/292.41 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.41 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.41 And the Tuples: 927.57/292.41 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.41 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.41 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.41 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.41 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.41 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.41 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.41 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.41 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.41 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.41 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.41 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.41 The order we found is given by the following interpretation: 927.57/292.41 927.57/292.41 Polynomial interpretation : 927.57/292.41 927.57/292.41 POL(0) = 0 927.57/292.41 POL(<=(x_1, x_2)) = [1] 927.57/292.41 POL(<='(x_1, x_2)) = 0 927.57/292.41 POL(Cons(x_1, x_2)) = [2] + x_2 927.57/292.41 POL(False) = [1] 927.57/292.41 POL(MERGE(x_1, x_2)) = x_2 927.57/292.41 POL(MERGESORT(x_1)) = x_1^2 927.57/292.41 POL(MERGE[ITE](x_1, x_2, x_3)) = x_1*x_3 927.57/292.41 POL(Nil) = 0 927.57/292.41 POL(S(x_1)) = 0 927.57/292.41 POL(SPLITMERGE(x_1, x_2, x_3)) = x_3 + x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 927.57/292.41 POL(True) = [1] 927.57/292.41 POL(c(x_1)) = x_1 927.57/292.41 POL(c11(x_1)) = x_1 927.57/292.41 POL(c12(x_1)) = x_1 927.57/292.41 POL(c12(x_1, x_2)) = x_1 + x_2 927.57/292.41 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.41 POL(c3(x_1)) = x_1 927.57/292.41 POL(c4(x_1)) = x_1 927.57/292.41 POL(c5(x_1)) = x_1 927.57/292.41 POL(c8(x_1)) = x_1 927.57/292.41 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(merge(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 927.57/292.42 POL(mergesort(x_1)) = x_1 927.57/292.42 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (34) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.42 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 We considered the (Usable) Rules: 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 And the Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 The order we found is given by the following interpretation: 927.57/292.42 927.57/292.42 Polynomial interpretation : 927.57/292.42 927.57/292.42 POL(0) = 0 927.57/292.42 POL(<=(x_1, x_2)) = [1] 927.57/292.42 POL(<='(x_1, x_2)) = 0 927.57/292.42 POL(Cons(x_1, x_2)) = [2] + x_2 927.57/292.42 POL(False) = [1] 927.57/292.42 POL(MERGE(x_1, x_2)) = x_2 + x_1*x_2 927.57/292.42 POL(MERGESORT(x_1)) = x_1^2 927.57/292.42 POL(MERGE[ITE](x_1, x_2, x_3)) = x_2*x_3 + x_1*x_3 927.57/292.42 POL(Nil) = 0 927.57/292.42 POL(S(x_1)) = 0 927.57/292.42 POL(SPLITMERGE(x_1, x_2, x_3)) = x_3 + x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 927.57/292.42 POL(True) = 0 927.57/292.42 POL(c(x_1)) = x_1 927.57/292.42 POL(c11(x_1)) = x_1 927.57/292.42 POL(c12(x_1)) = x_1 927.57/292.42 POL(c12(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.42 POL(c3(x_1)) = x_1 927.57/292.42 POL(c4(x_1)) = x_1 927.57/292.42 POL(c5(x_1)) = x_1 927.57/292.42 POL(c8(x_1)) = x_1 927.57/292.42 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(merge(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 927.57/292.42 POL(mergesort(x_1)) = x_1 927.57/292.42 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (36) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (37) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.42 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 We considered the (Usable) Rules: 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 And the Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 The order we found is given by the following interpretation: 927.57/292.42 927.57/292.42 Polynomial interpretation : 927.57/292.42 927.57/292.42 POL(0) = 0 927.57/292.42 POL(<=(x_1, x_2)) = [1] 927.57/292.42 POL(<='(x_1, x_2)) = 0 927.57/292.42 POL(Cons(x_1, x_2)) = [1] + x_2 927.57/292.42 POL(False) = 0 927.57/292.42 POL(MERGE(x_1, x_2)) = [1] + x_2 927.57/292.42 POL(MERGESORT(x_1)) = [2]x_1^2 927.57/292.42 POL(MERGE[ITE](x_1, x_2, x_3)) = x_1 + x_3 927.57/292.42 POL(Nil) = 0 927.57/292.42 POL(S(x_1)) = 0 927.57/292.42 POL(SPLITMERGE(x_1, x_2, x_3)) = [2] + x_1 + [2]x_3 + [2]x_3^2 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + [2]x_2^2 927.57/292.42 POL(True) = [1] 927.57/292.42 POL(c(x_1)) = x_1 927.57/292.42 POL(c11(x_1)) = x_1 927.57/292.42 POL(c12(x_1)) = x_1 927.57/292.42 POL(c12(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.42 POL(c3(x_1)) = x_1 927.57/292.42 POL(c4(x_1)) = x_1 927.57/292.42 POL(c5(x_1)) = x_1 927.57/292.42 POL(c8(x_1)) = x_1 927.57/292.42 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(merge(x_1, x_2)) = x_1 + x_2 927.57/292.42 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 927.57/292.42 POL(mergesort(x_1)) = [2]x_1 927.57/292.42 POL(splitmerge(x_1, x_2, x_3)) = [2]x_1 + [2]x_2 + [2]x_3 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (38) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(z1, z2))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(z1, z2)))) by 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (40) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGE(Nil, splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Nil), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Removed 3 trailing tuple parts 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (42) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) 927.57/292.42 Removed 1 leading nodes: 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (44) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_3, c12_2, c12_1, c8_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), x1) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), mergesort(x1)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(x1)) by 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))), MERGESORT(Cons(z0, Nil))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Cons(x0, Cons(x1, x2))), MERGESORT(Nil)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (46) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))), MERGESORT(Cons(z0, Nil))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Nil), MERGESORT(Cons(x0, Cons(x1, x2))), MERGESORT(Nil)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c8_2, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Removed 3 trailing tuple parts 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (48) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_2, c12_1, c8_2, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Use narrowing to replace SPLITMERGE(Nil, x0, Cons(z0, Nil)) -> c12(MERGE(mergesort(x0), Cons(z0, Nil)), MERGESORT(x0)) by 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil))) 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil)) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (50) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil))) 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil)) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c12_2, c8_2, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Removed 1 trailing nodes: 927.57/292.42 SPLITMERGE(Nil, Nil, Cons(x1, Nil)) -> c12(MERGE(Nil, Cons(x1, Nil)), MERGESORT(Nil)) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (52) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Nil))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c12_2, c8_2, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Removed 1 trailing tuple parts 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (54) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c12_2, c8_2, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Use narrowing to replace SPLITMERGE(Nil, Cons(z0, Nil), x1) -> c12(MERGE(Cons(z0, Nil), mergesort(x1)), MERGESORT(x1)) by 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c12(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil)) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (56) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c12(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil)) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Removed 1 trailing nodes: 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Nil) -> c12(MERGE(Cons(x0, Nil), Nil), MERGESORT(Nil)) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (58) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Nil)) -> c12(MERGE(Cons(x0, Nil), Cons(z0, Nil)), MERGESORT(Cons(z0, Nil))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Removed 1 trailing tuple parts 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (60) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c3_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (61) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Use instantiation to replace MERGE[ITE](False, z0, Cons(z1, z2)) -> c3(MERGE(z0, z2)) by 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (62) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2, c3_1 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (63) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, x0, Cons(z0, Cons(x2, x3))) -> c12(MERGE(mergesort(x0), splitmerge(Cons(x2, x3), Cons(z0, Nil), Nil)), MERGESORT(x0), MERGESORT(Cons(z0, Cons(x2, x3)))) by SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (64) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2, c3_1 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (66) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2, c3_1 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (67) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (68) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Cons(x2, x3))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_3, c12_2, c3_1 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Cons(z1, z2))) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(x0, Cons(x1, x2))), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (70) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c12_3, c3_1 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(x1, x2)), x3) -> c12(MERGE(splitmerge(Cons(x1, x2), Cons(z0, Nil), Nil), mergesort(x3)), MERGESORT(Cons(z0, Cons(x1, x2))), MERGESORT(x3)) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (72) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c3_1, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (74) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Cons(z0, Nil)) -> c12(MERGE(splitmerge(Cons(x0, Cons(x1, x2)), Nil, Nil), Cons(z0, Nil)), MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c3_1, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(x1, Nil)) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), Cons(x1, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (76) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Cons(x2, x3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(x1, Cons(x2, x3)), Nil, Nil)), MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c12_2, c3_1, c12_3 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(x0, Nil), Cons(z0, Cons(z1, z2))) -> c12(MERGE(Cons(x0, Nil), splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (78) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (80) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (82) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(Cons(z4, z5), Cons(z3, Nil), Nil)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (84) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (86) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.42 927.57/292.42 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.42 927.57/292.42 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.42 927.57/292.42 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.42 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(Cons(z1, z2), Cons(z0, Nil), Nil), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) by SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 ---------------------------------------- 927.57/292.42 927.57/292.42 (88) 927.57/292.42 Obligation: 927.57/292.42 Complexity Dependency Tuples Problem 927.57/292.42 927.57/292.42 Rules: 927.57/292.42 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.42 <=(0, z0) -> True 927.57/292.42 <=(S(z0), 0) -> False 927.57/292.42 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.42 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.42 mergesort(Nil) -> Nil 927.57/292.42 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.42 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.42 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.42 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.42 merge(Nil, z0) -> z0 927.57/292.42 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.42 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.42 Tuples: 927.57/292.42 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.42 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.42 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.42 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.42 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.42 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.42 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.42 S tuples: 927.57/292.42 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.42 K tuples: 927.57/292.42 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.42 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.42 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.42 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.42 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.43 927.57/292.43 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Used rewriting to replace SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(Cons(z2, z3), Cons(z1, Nil), Nil)), MERGESORT(Cons(z1, Cons(z2, z3)))) by SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (90) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.43 MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: <='_2, MERGE[ITE]_3, MERGESORT_1, SPLITMERGE_3, MERGE_2 927.57/292.43 927.57/292.43 Compound Symbols: c_1, c4_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (91) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace MERGE[ITE](True, Cons(z0, z1), z2) -> c4(MERGE(z1, z2)) by 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (92) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 <='(S(z0), S(z1)) -> c(<='(z0, z1)) 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: <='_2, MERGESORT_1, SPLITMERGE_3, MERGE_2, MERGE[ITE]_3 927.57/292.43 927.57/292.43 Compound Symbols: c_1, c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2, c4_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use forward instantiation to replace <='(S(z0), S(z1)) -> c(<='(z0, z1)) by 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (94) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3)), <='(S(0), S(z0))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3)), <='(S(S(z0)), S(0))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, SPLITMERGE_3, MERGE_2, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2, c4_1, c_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Removed 2 trailing tuple parts 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (96) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, SPLITMERGE_3, MERGE_2, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c11_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2, c4_1, c_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Cons(z0, z1), z2, z3) -> c11(SPLITMERGE(z1, Cons(z0, z3), z2)) by 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (98) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c12_1, c8_2, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (99) CdtLeafRemovalProof (ComplexityIfPolyImplication) 927.57/292.43 Removed 2 leading nodes: 927.57/292.43 SPLITMERGE(Nil, x0, Nil) -> c12(MERGESORT(x0)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), Nil) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (100) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Nil, x0, Cons(x1, Cons(x2, x3))) -> c12(MERGESORT(Cons(x1, Cons(x2, x3)))) by 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (102) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (103) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Nil, Cons(x0, Cons(x1, x2)), x3) -> c12(MERGESORT(Cons(x0, Cons(x1, x2)))) by 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (104) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use forward instantiation to replace MERGE(Cons(S(x0), x1), Cons(S(x2), x3)) -> c8(<='(S(x0), S(x2))) by 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (106) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (107) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Nil, z0, Cons(z1, Cons(z2, z3))) -> c12(MERGE(mergesort(z0), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(z0), MERGESORT(Cons(z1, Cons(z2, z3)))) by 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (108) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (109) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), z3) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), mergesort(z3)), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(z3)) by 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (110) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (111) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Cons(z0, z1), Cons(x0, x3), x2) -> c11(SPLITMERGE(z1, Cons(z0, x2), Cons(x0, x3))) by 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (112) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c12(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil))), MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (113) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Split RHS of tuples not part of any SCC 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (114) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1, c1_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (115) CdtLeafRemovalProof (ComplexityIfPolyImplication) 927.57/292.43 Removed 1 leading nodes: 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGESORT(Cons(z1, Cons(z2, z3)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (116) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1, c1_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (117) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGESORT(Cons(x2, Cons(z2, z3)))) by 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (118) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1, c1_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (119) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 Use instantiation to replace SPLITMERGE(Nil, Cons(x0, x4), Cons(x2, Cons(z2, z3))) -> c12(MERGE(mergesort(Cons(x0, x4)), splitmerge(z3, Cons(z2, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, x4)), MERGESORT(Cons(x2, Cons(z2, z3)))) by 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(z4, Cons(z3, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, Cons(x4, x5))), MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (120) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(z4, Cons(z3, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, Cons(x4, x5))), MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 S tuples: 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1, c1_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 927.57/292.43 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 We considered the (Usable) Rules: 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 And the Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(z4, Cons(z3, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, Cons(x4, x5))), MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 The order we found is given by the following interpretation: 927.57/292.43 927.57/292.43 Polynomial interpretation : 927.57/292.43 927.57/292.43 POL(0) = 0 927.57/292.43 POL(<=(x_1, x_2)) = 0 927.57/292.43 POL(<='(x_1, x_2)) = 0 927.57/292.43 POL(Cons(x_1, x_2)) = [2] + x_2 927.57/292.43 POL(False) = 0 927.57/292.43 POL(MERGE(x_1, x_2)) = [2] + x_1 + x_1*x_2 927.57/292.43 POL(MERGESORT(x_1)) = [1] + x_1^2 927.57/292.43 POL(MERGE[ITE](x_1, x_2, x_3)) = x_2 + x_2*x_3 927.57/292.43 POL(Nil) = 0 927.57/292.43 POL(S(x_1)) = 0 927.57/292.43 POL(SPLITMERGE(x_1, x_2, x_3)) = x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 + x_1^2 + [2]x_1*x_2 + x_2^2 927.57/292.43 POL(True) = 0 927.57/292.43 POL(c(x_1)) = x_1 927.57/292.43 POL(c1(x_1)) = x_1 927.57/292.43 POL(c11(x_1)) = x_1 927.57/292.43 POL(c12(x_1)) = x_1 927.57/292.43 POL(c12(x_1, x_2)) = x_1 + x_2 927.57/292.43 POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.43 POL(c3(x_1)) = x_1 927.57/292.43 POL(c4(x_1)) = x_1 927.57/292.43 POL(c5(x_1)) = x_1 927.57/292.43 POL(c8(x_1)) = x_1 927.57/292.43 POL(c8(x_1, x_2)) = x_1 + x_2 927.57/292.43 POL(merge(x_1, x_2)) = x_1 + x_2 927.57/292.43 POL(merge[Ite](x_1, x_2, x_3)) = x_2 + x_3 927.57/292.43 POL(mergesort(x_1)) = x_1 927.57/292.43 POL(splitmerge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (122) 927.57/292.43 Obligation: 927.57/292.43 Complexity Dependency Tuples Problem 927.57/292.43 927.57/292.43 Rules: 927.57/292.43 <=(S(z0), S(z1)) -> <=(z0, z1) 927.57/292.43 <=(0, z0) -> True 927.57/292.43 <=(S(z0), 0) -> False 927.57/292.43 mergesort(Cons(z0, Cons(z1, z2))) -> splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil) 927.57/292.43 mergesort(Cons(z0, Nil)) -> Cons(z0, Nil) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 splitmerge(Cons(z0, z1), z2, z3) -> splitmerge(z1, Cons(z0, z3), z2) 927.57/292.43 splitmerge(Nil, z0, z1) -> merge(mergesort(z0), mergesort(z1)) 927.57/292.43 merge(Cons(z0, z1), Cons(z2, z3)) -> merge[Ite](<=(z0, z2), Cons(z0, z1), Cons(z2, z3)) 927.57/292.43 merge(Cons(z0, z1), Nil) -> Cons(z0, z1) 927.57/292.43 merge(Nil, z0) -> z0 927.57/292.43 merge[Ite](False, z0, Cons(z1, z2)) -> Cons(z1, merge(z0, z2)) 927.57/292.43 merge[Ite](True, Cons(z0, z1), z2) -> Cons(z0, merge(z1, z2)) 927.57/292.43 Tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(x1, Nil)) -> c12(MERGE(Cons(z0, Nil), Cons(x1, Nil))) 927.57/292.43 MERGE[ITE](False, Cons(S(x0), x1), Cons(0, x2)) -> c3(MERGE(Cons(S(x0), x1), x2)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c3(MERGE(Cons(S(S(x0)), x1), x3)) 927.57/292.43 MERGE[ITE](False, Cons(S(S(x0)), x1), Cons(S(0), x2)) -> c3(MERGE(Cons(S(S(x0)), x1), x2)) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Cons(z4, z5))) -> c12(MERGE(splitmerge(Cons(z0, Cons(z1, z2)), Nil, Nil), splitmerge(z5, Cons(z4, Nil), Cons(z3, Nil))), MERGESORT(Cons(z0, Cons(z1, z2))), MERGESORT(Cons(z3, Cons(z4, z5)))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(z0, Nil)), Cons(z3, Nil)), MERGESORT(Cons(z0, Cons(z1, z2)))) 927.57/292.43 MERGE[ITE](True, Cons(0, x0), Cons(x1, x2)) -> c4(MERGE(x0, Cons(x1, x2))) 927.57/292.43 MERGE[ITE](True, Cons(S(S(x0)), x1), Cons(S(S(x2)), x3)) -> c4(MERGE(x1, Cons(S(S(x2)), x3))) 927.57/292.43 MERGE[ITE](True, Cons(S(0), x0), Cons(S(x1), x2)) -> c4(MERGE(x0, Cons(S(x1), x2))) 927.57/292.43 <='(S(S(y0)), S(S(y1))) -> c(<='(S(y0), S(y1))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGESORT(Cons(x0, Cons(z1, z2)))) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(z1, z2)), Cons(x2, x3)) -> c12(MERGE(splitmerge(z2, Cons(z1, Nil), Cons(x0, Nil)), mergesort(Cons(x2, x3))), MERGESORT(Cons(x0, Cons(z1, z2))), MERGESORT(Cons(x2, x3))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 SPLITMERGE(Nil, Cons(z0, Nil), Cons(z1, Cons(z2, z3))) -> c1(MERGE(Cons(z0, Nil), splitmerge(z3, Cons(z2, Nil), Cons(z1, Nil)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 SPLITMERGE(Nil, Cons(x0, Cons(x4, x5)), Cons(x2, Cons(z3, z4))) -> c12(MERGE(mergesort(Cons(x0, Cons(x4, x5))), splitmerge(z4, Cons(z3, Nil), Cons(x2, Nil))), MERGESORT(Cons(x0, Cons(x4, x5))), MERGESORT(Cons(x2, Cons(z3, z4)))) 927.57/292.43 S tuples:none 927.57/292.43 K tuples: 927.57/292.43 MERGESORT(Cons(z0, Cons(z1, z2))) -> c5(SPLITMERGE(Cons(z0, Cons(z1, z2)), Nil, Nil)) 927.57/292.43 MERGE(Cons(0, x1), Cons(z0, x3)) -> c8(MERGE[ITE](True, Cons(0, x1), Cons(z0, x3))) 927.57/292.43 MERGE(Cons(S(z0), x1), Cons(0, x3)) -> c8(MERGE[ITE](False, Cons(S(z0), x1), Cons(0, x3))) 927.57/292.43 MERGE(Cons(S(0), x1), Cons(S(z0), x3)) -> c8(MERGE[ITE](True, Cons(S(0), x1), Cons(S(z0), x3))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(0), x3)) -> c8(MERGE[ITE](False, Cons(S(S(z0)), x1), Cons(S(0), x3))) 927.57/292.43 SPLITMERGE(Cons(x0, Cons(x1, x2)), Nil, Nil) -> c11(SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil)) 927.57/292.43 MERGE(Cons(S(S(y0)), z1), Cons(S(S(y1)), z3)) -> c8(<='(S(S(y0)), S(S(y1)))) 927.57/292.43 SPLITMERGE(Cons(x1, x2), Cons(x0, Nil), Nil) -> c11(SPLITMERGE(x2, Cons(x1, Nil), Cons(x0, Nil))) 927.57/292.43 SPLITMERGE(Cons(z0, z1), Cons(x0, x4), Cons(x2, x3)) -> c11(SPLITMERGE(z1, Cons(z0, Cons(x2, x3)), Cons(x0, x4))) 927.57/292.43 MERGE(Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)) -> c8(MERGE[ITE](<=(z0, z1), Cons(S(S(z0)), x1), Cons(S(S(z1)), x3)), <='(S(S(z0)), S(S(z1)))) 927.57/292.43 Defined Rule Symbols: <=_2, mergesort_1, splitmerge_3, merge_2, merge[Ite]_3 927.57/292.43 927.57/292.43 Defined Pair Symbols: MERGESORT_1, MERGE_2, SPLITMERGE_3, MERGE[ITE]_3, <='_2 927.57/292.43 927.57/292.43 Compound Symbols: c5_1, c8_1, c8_2, c12_1, c3_1, c12_3, c12_2, c4_1, c_1, c11_1, c1_1 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (123) SIsEmptyProof (BOTH BOUNDS(ID, ID)) 927.57/292.43 The set S is empty 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (124) 927.57/292.43 BOUNDS(1, 1) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (125) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 927.57/292.43 Transformed a relative TRS into a decreasing-loop problem. 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (126) 927.57/292.43 Obligation: 927.57/292.43 Analyzing the following TRS for decreasing loops: 927.57/292.43 927.57/292.43 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 927.57/292.43 927.57/292.43 927.57/292.43 The TRS R consists of the following rules: 927.57/292.43 927.57/292.43 mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) 927.57/292.43 mergesort(Cons(x, Nil)) -> Cons(x, Nil) 927.57/292.43 merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) 927.57/292.43 merge(Cons(x, xs), Nil) -> Cons(x, xs) 927.57/292.43 splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) 927.57/292.43 splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 merge(Nil, xs2) -> xs2 927.57/292.43 notEmpty(Cons(x, xs)) -> True 927.57/292.43 notEmpty(Nil) -> False 927.57/292.43 goal(xs) -> mergesort(xs) 927.57/292.43 927.57/292.43 The (relative) TRS S consists of the following rules: 927.57/292.43 927.57/292.43 <=(S(x), S(y)) -> <=(x, y) 927.57/292.43 <=(0, y) -> True 927.57/292.43 <=(S(x), 0) -> False 927.57/292.43 merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) 927.57/292.43 merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) 927.57/292.43 927.57/292.43 Rewrite Strategy: INNERMOST 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (127) DecreasingLoopProof (LOWER BOUND(ID)) 927.57/292.43 The following loop(s) give(s) rise to the lower bound Omega(n^1): 927.57/292.43 927.57/292.43 The rewrite sequence 927.57/292.43 927.57/292.43 splitmerge(Cons(x, xs), xs1, xs2) ->^+ splitmerge(xs, Cons(x, xs2), xs1) 927.57/292.43 927.57/292.43 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 927.57/292.43 927.57/292.43 The pumping substitution is [xs / Cons(x, xs)]. 927.57/292.43 927.57/292.43 The result substitution is [xs1 / Cons(x, xs2), xs2 / xs1]. 927.57/292.43 927.57/292.43 927.57/292.43 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (128) 927.57/292.43 Complex Obligation (BEST) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (129) 927.57/292.43 Obligation: 927.57/292.43 Proved the lower bound n^1 for the following obligation: 927.57/292.43 927.57/292.43 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 927.57/292.43 927.57/292.43 927.57/292.43 The TRS R consists of the following rules: 927.57/292.43 927.57/292.43 mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) 927.57/292.43 mergesort(Cons(x, Nil)) -> Cons(x, Nil) 927.57/292.43 merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) 927.57/292.43 merge(Cons(x, xs), Nil) -> Cons(x, xs) 927.57/292.43 splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) 927.57/292.43 splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 merge(Nil, xs2) -> xs2 927.57/292.43 notEmpty(Cons(x, xs)) -> True 927.57/292.43 notEmpty(Nil) -> False 927.57/292.43 goal(xs) -> mergesort(xs) 927.57/292.43 927.57/292.43 The (relative) TRS S consists of the following rules: 927.57/292.43 927.57/292.43 <=(S(x), S(y)) -> <=(x, y) 927.57/292.43 <=(0, y) -> True 927.57/292.43 <=(S(x), 0) -> False 927.57/292.43 merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) 927.57/292.43 merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) 927.57/292.43 927.57/292.43 Rewrite Strategy: INNERMOST 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (130) LowerBoundPropagationProof (FINISHED) 927.57/292.43 Propagated lower bound. 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (131) 927.57/292.43 BOUNDS(n^1, INF) 927.57/292.43 927.57/292.43 ---------------------------------------- 927.57/292.43 927.57/292.43 (132) 927.57/292.43 Obligation: 927.57/292.43 Analyzing the following TRS for decreasing loops: 927.57/292.43 927.57/292.43 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 927.57/292.43 927.57/292.43 927.57/292.43 The TRS R consists of the following rules: 927.57/292.43 927.57/292.43 mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil) 927.57/292.43 mergesort(Cons(x, Nil)) -> Cons(x, Nil) 927.57/292.43 merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) 927.57/292.43 merge(Cons(x, xs), Nil) -> Cons(x, xs) 927.57/292.43 splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1) 927.57/292.43 splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2)) 927.57/292.43 mergesort(Nil) -> Nil 927.57/292.43 merge(Nil, xs2) -> xs2 927.57/292.43 notEmpty(Cons(x, xs)) -> True 927.57/292.43 notEmpty(Nil) -> False 927.57/292.43 goal(xs) -> mergesort(xs) 927.57/292.43 927.57/292.43 The (relative) TRS S consists of the following rules: 927.57/292.43 927.57/292.43 <=(S(x), S(y)) -> <=(x, y) 927.57/292.43 <=(0, y) -> True 927.57/292.43 <=(S(x), 0) -> False 927.57/292.43 merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs)) 927.57/292.43 merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2)) 927.57/292.43 927.57/292.43 Rewrite Strategy: INNERMOST 927.69/292.48 EOF