KILLED proof of input_2JnhxNTcqt.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 540 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 27 ms] (48) CdtProblem (49) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 446 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 799 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 933 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 965 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 982 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 961 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 2 ms] (108) CdtProblem (109) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 926 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 609 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1307 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 907 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true function(iszero, s(x), dummy, dummy2) -> false function(p, 0, dummy, dummy2) -> 0 function(p, s(0), dummy, dummy2) -> 0 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) function(if, true, x, y) -> y function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) function(third, x, y, z) -> z S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: function(iszero, 0', dummy, dummy2) -> true function(iszero, s(x), dummy, dummy2) -> false function(p, 0', dummy, dummy2) -> 0' function(p, s(0'), dummy, dummy2) -> 0' function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) function(if, true, x, y) -> y function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) function(third, x, y, z) -> z S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true function(iszero, s(x), dummy, dummy2) -> false function(p, 0, dummy, dummy2) -> 0 function(p, s(0), dummy, dummy2) -> 0 function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) function(if, true, x, y) -> y function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) function(third, x, y, z) -> z S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false -> 0:true:s:false -> 0:true:s:false -> 0:true:s:false iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false true :: 0:true:s:false s :: 0:true:s:false -> 0:true:s:false false :: 0:true:s:false p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: function_4 Due to the following rules being added: function(v0, v1, v2, v3) -> null_function [0] And the following fresh constants: null_function ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false:null_function true :: 0:true:s:false:null_function s :: 0:true:s:false:null_function -> 0:true:s:false:null_function false :: 0:true:s:false:null_function p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third null_function :: 0:true:s:false:null_function Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, 0, y) -> function(if, true, 0, y) [2] function(plus, dummy, s(x'), y) -> function(if, false, s(x'), y) [2] function(plus, dummy, x, y) -> function(if, null_function, x, y) [1] function(if, true, x, y) -> y [1] function(if, false, 0, y) -> function(plus, y, 0, s(y)) [3] function(if, false, s(0), y) -> function(plus, y, 0, s(y)) [3] function(if, false, s(s(x'')), y) -> function(plus, y, s(function(p, s(x''), x'', x'')), s(y)) [3] function(if, false, x, y) -> function(plus, y, null_function, s(y)) [2] function(if, false, 0, y) -> function(plus, null_function, 0, s(y)) [2] function(if, false, s(0), y) -> function(plus, null_function, 0, s(y)) [2] function(if, false, s(s(x1)), y) -> function(plus, null_function, s(function(p, s(x1), x1, x1)), s(y)) [2] function(if, false, x, y) -> function(plus, null_function, null_function, s(y)) [1] function(third, x, y, z) -> z [1] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false:null_function true :: 0:true:s:false:null_function s :: 0:true:s:false:null_function -> 0:true:s:false:null_function false :: 0:true:s:false:null_function p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third null_function :: 0:true:s:false:null_function Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: iszero => 1 0 => 0 true => 2 false => 1 p => 2 plus => 3 if => 0 third => 4 null_function => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: function(z', z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 1 }-> z :|: z' = 4, z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x function(z', z'', z1, z2) -{ 3 }-> function(3, y, 0, 1 + y) :|: z1 = 0, z2 = y, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, y, 0, 1 + y) :|: z2 = y, z1 = 1 + 0, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, y, 0, 1 + y) :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, y, 1 + function(2, 1 + x'', x'', x''), 1 + y) :|: z2 = y, y >= 0, x'' >= 0, z'' = 1, z1 = 1 + (1 + x''), z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + y) :|: z1 = 0, z2 = y, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + y) :|: z2 = y, z1 = 1 + 0, y >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 1 }-> function(3, 0, 0, 1 + y) :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 1 + function(2, 1 + x1, x1, x1), 1 + y) :|: z2 = y, x1 >= 0, y >= 0, z'' = 1, z' = 0, z1 = 1 + (1 + x1) function(z', z'', z1, z2) -{ 2 }-> function(0, 2, 0, y) :|: z' = 3, z1 = 0, z2 = y, dummy >= 0, y >= 0, z'' = dummy function(z', z'', z1, z2) -{ 2 }-> function(0, 1, 1 + x', y) :|: z' = 3, z2 = y, dummy >= 0, x' >= 0, y >= 0, z'' = dummy, z1 = 1 + x' function(z', z'', z1, z2) -{ 1 }-> function(0, 0, x, y) :|: z' = 3, z2 = y, dummy >= 0, x >= 0, y >= 0, z'' = dummy, z1 = x function(z', z'', z1, z2) -{ 1 }-> 2 :|: z'' = 0, dummy2 >= 0, z2 = dummy2, dummy >= 0, z' = 1, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 1 :|: dummy2 >= 0, z2 = dummy2, x >= 0, dummy >= 0, z' = 1, z'' = 1 + x, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z'' = 1 + 0, z1 = dummy function(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 function(z', z'', z1, z2) -{ 1 }-> 1 + function(2, 1 + x, x, x) :|: dummy2 >= 0, z' = 2, z2 = dummy2, x >= 0, dummy >= 0, z'' = 1 + (1 + x), z1 = dummy ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: function(z', z'', z1, z2) -{ 1 }-> z2 :|: z1 >= 0, z2 >= 0, z'' = 2, z' = 0 function(z', z'', z1, z2) -{ 1 }-> z2 :|: z' = 4, z2 >= 0, z'' >= 0, z1 >= 0 function(z', z'', z1, z2) -{ 3 }-> function(3, z2, 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, z2, 0, 1 + z2) :|: z1 = 1 + 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, z2, 0, 1 + z2) :|: z1 >= 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 3 }-> function(3, z2, 1 + function(2, 1 + (z1 - 2), z1 - 2, z1 - 2), 1 + z2) :|: z2 >= 0, z1 - 2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 0, 1 + z2) :|: z1 = 1 + 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 1 }-> function(3, 0, 0, 1 + z2) :|: z1 >= 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(3, 0, 1 + function(2, 1 + (z1 - 2), z1 - 2, z1 - 2), 1 + z2) :|: z1 - 2 >= 0, z2 >= 0, z'' = 1, z' = 0 function(z', z'', z1, z2) -{ 2 }-> function(0, 2, 0, z2) :|: z' = 3, z1 = 0, z'' >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 2 }-> function(0, 1, 1 + (z1 - 1), z2) :|: z' = 3, z'' >= 0, z1 - 1 >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 1 }-> function(0, 0, z1, z2) :|: z' = 3, z'' >= 0, z1 >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 1 }-> 2 :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 function(z', z'', z1, z2) -{ 1 }-> 1 :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1 function(z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z2 >= 0, z' = 2, z1 >= 0 function(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z1 >= 0, z'' = 1 + 0 function(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 function(z', z'', z1, z2) -{ 1 }-> 1 + function(2, 1 + (z'' - 2), z'' - 2, z'' - 2) :|: z2 >= 0, z' = 2, z'' - 2 >= 0, z1 >= 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: function(v0, v1, v2, v3) -> null_function [0] And the following fresh constants: null_function ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: function(iszero, 0, dummy, dummy2) -> true [1] function(iszero, s(x), dummy, dummy2) -> false [1] function(p, 0, dummy, dummy2) -> 0 [1] function(p, s(0), dummy, dummy2) -> 0 [1] function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x)) [1] function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y) [1] function(if, true, x, y) -> y [1] function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y)) [1] function(third, x, y, z) -> z [1] function(v0, v1, v2, v3) -> null_function [0] The TRS has the following type information: function :: iszero:p:plus:if:third -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function -> 0:true:s:false:null_function iszero :: iszero:p:plus:if:third 0 :: 0:true:s:false:null_function true :: 0:true:s:false:null_function s :: 0:true:s:false:null_function -> 0:true:s:false:null_function false :: 0:true:s:false:null_function p :: iszero:p:plus:if:third plus :: iszero:p:plus:if:third if :: iszero:p:plus:if:third third :: iszero:p:plus:if:third null_function :: 0:true:s:false:null_function Rewrite Strategy: INNERMOST ---------------------------------------- (19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: iszero => 1 0 => 0 true => 2 false => 1 p => 2 plus => 3 if => 0 third => 4 null_function => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: function(z', z'', z1, z2) -{ 1 }-> y :|: z2 = y, x >= 0, y >= 0, z'' = 2, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 1 }-> z :|: z' = 4, z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x function(z', z'', z1, z2) -{ 1 }-> function(3, function(4, x, y, y), function(2, x, x, y), 1 + y) :|: z2 = y, x >= 0, y >= 0, z'' = 1, z1 = x, z' = 0 function(z', z'', z1, z2) -{ 1 }-> function(0, function(1, x, x, x), x, y) :|: z' = 3, z2 = y, dummy >= 0, x >= 0, y >= 0, z'' = dummy, z1 = x function(z', z'', z1, z2) -{ 1 }-> 2 :|: z'' = 0, dummy2 >= 0, z2 = dummy2, dummy >= 0, z' = 1, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 1 :|: dummy2 >= 0, z2 = dummy2, x >= 0, dummy >= 0, z' = 1, z'' = 1 + x, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z1 = dummy function(z', z'', z1, z2) -{ 1 }-> 0 :|: dummy2 >= 0, z' = 2, z2 = dummy2, dummy >= 0, z'' = 1 + 0, z1 = dummy function(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 function(z', z'', z1, z2) -{ 1 }-> 1 + function(2, 1 + x, x, x) :|: dummy2 >= 0, z' = 2, z2 = dummy2, x >= 0, dummy >= 0, z'' = 1 + (1 + x), z1 = dummy Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 S tuples: FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, true, z0, z1) -> c6 FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(third, z0, z1, z2) -> c9 K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c, c1, c2, c3, c4_1, c5_2, c6, c7_2, c8_2, c9 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: FUNCTION(if, true, z0, z1) -> c6 FUNCTION(p, 0, z0, z1) -> c2 FUNCTION(iszero, s(z0), z1, z2) -> c1 FUNCTION(p, s(0), z0, z1) -> c3 FUNCTION(iszero, 0, z0, z1) -> c FUNCTION(third, z0, z1, z2) -> c9 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_2, c7_2, c8_2 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) function(third, z0, z1, z2) -> z2 Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (27) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2) function(if, true, z0, z1) -> z1 function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) by FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) We considered the (Usable) Rules: function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(iszero) = 0 POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [1] POL(third) = 0 POL(true) = [1] ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) K tuples: FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2)) by FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) K tuples: FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c7_1, c8_2, c8_1, c5_1 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: FUNCTION(plus, x0, 0, x2) -> c5(FUNCTION(if, true, 0, x2)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c7_1, c8_2, c5_1 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(iszero, 0, z0, z1) -> true function(iszero, s(z0), z1, z2) -> false ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c7_1, c8_2, c5_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))) by FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, function(third, 0, z1, z1), 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, function(third, s(0), z1, z1), 0, s(z1))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) by FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) K tuples:none Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1, c8_1 ---------------------------------------- (47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) We considered the (Usable) Rules:none And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1, c8_1 ---------------------------------------- (49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) We considered the (Usable) Rules: function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c8_2, c5_1, c7_1, c8_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1)) by FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, s(s(z0)), z2) -> c8(FUNCTION(plus, z2, s(function(p, s(z0), z0, z0)), s(z2)), FUNCTION(p, s(s(z0)), s(s(z0)), z2)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FUNCTION(if, false, s(0), z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, s(0), s(0), z1)) FUNCTION(if, false, 0, z1) -> c8(FUNCTION(plus, z1, 0, s(z1)), FUNCTION(p, 0, 0, z1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, function(third, s(s(z0)), z2, z2), s(function(p, s(z0), z0, z0)), s(z2))) by FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) We considered the (Usable) Rules: function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [2] + x_1 POL(third) = 0 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c7_1, c8_2, c8_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, z1, function(p, z0, z0, z1), s(z1))) by FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(s(z0)), z2) -> c7(FUNCTION(plus, z2, s(function(p, s(z0), z0, z0)), s(z2))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FUNCTION(if, false, s(0), z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) FUNCTION(if, false, 0, z1) -> c7(FUNCTION(plus, z1, 0, s(z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, 0, z0, z1) -> 0 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (63) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(p, 0, z0, z1) -> 0 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) by FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(z0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1)), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), x1)) by FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) We considered the (Usable) Rules:none And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + [3]x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [3] + x_1 POL(third) = 0 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [2] + x_1 POL(third) = 0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) by FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(z0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(z0))), x1, x1), s(s(function(p, s(z0), z0, z0))), s(x1))) by FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) by FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1)), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), x1)) by FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) We considered the (Usable) Rules:none And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [2]x_2 + [3]x_3 + [3]x_4 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [1] POL(third) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [2] + x_1 POL(third) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) by FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(z0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(z0)))), x1, x1), s(s(s(function(p, s(z0), z0, z0)))), s(x1))) by FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 POL(third) = 0 ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) by FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) S tuples: FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c4_1, c5_1, c8_2, c8_1, c7_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0)) by FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c8(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1)), FUNCTION(p, s(s(0)), s(s(0)), x1)) FUNCTION(if, false, s(s(0)), x1) -> c7(FUNCTION(plus, function(third, s(s(0)), x1, x1), s(0), s(x1))) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), x1)) FUNCTION(if, false, s(s(s(0))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(0))), x1, x1), s(s(0)), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FUNCTION(if, false, s(s(0)), x0) -> c8(FUNCTION(p, s(s(0)), s(s(0)), x0)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1)), FUNCTION(p, s(s(0)), s(s(0)), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c8(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), x1)) by FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) by FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FUNCTION(if, false, s(s(s(s(s(z0))))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(s(z0))))), x1, x1), s(s(s(s(function(p, s(z0), z0, z0))))), s(x1))) by FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: function(third, z0, z1, z2) -> z2 function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (107) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: function(third, z0, z1, z2) -> z2 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) S tuples: FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c5_1, c8_2, c8_1, c7_1, c4_1 ---------------------------------------- (109) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(plus, x0, s(z0), x2) -> c5(FUNCTION(if, false, s(z0), x2)) by FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5(FUNCTION(if, false, s(0), s(x0))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5(FUNCTION(if, false, s(0), s(x0))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5(FUNCTION(if, false, s(0), s(x0))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1, c5 ---------------------------------------- (113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(plus, x0, s(0), s(x0)) -> c5 We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5) = 0 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c4_1, c5_1, c5 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FUNCTION(if, false, s(s(x0)), x1) -> c8(FUNCTION(p, s(s(x0)), s(s(x0)), x1)) by FUNCTION(if, false, s(s(s(y0))), z1) -> c8(FUNCTION(p, s(s(s(y0))), s(s(s(y0))), z1)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(s(0)))), x1) -> c7(FUNCTION(plus, function(third, s(s(s(s(0)))), x1, x1), s(s(s(0))), s(x1))) FUNCTION(plus, x0, s(0), s(x0)) -> c5 Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c4_1, c5_1, c5 ---------------------------------------- (117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FUNCTION(plus, x0, s(0), s(x0)) -> c5 ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c4_1, c5_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FUNCTION(p, s(s(s(y0))), z1, z2) -> c4(FUNCTION(p, s(s(y0)), s(y0), s(y0))) by FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c5_1, c4_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FUNCTION(if, false, s(s(s(0))), x0) -> c8(FUNCTION(p, s(s(s(0))), s(s(s(0))), x0)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1)), FUNCTION(p, s(s(s(0))), s(s(s(0))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c5_1, c4_1 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c7_1, c8_1, c5_1, c4_1 ---------------------------------------- (125) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(x0)), z1) -> c8(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1)), FUNCTION(p, s(s(x0)), s(s(x0)), z1)) by FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0))), FUNCTION(p, s(s(0)), s(s(0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0))), FUNCTION(p, s(s(0)), s(s(0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0))), FUNCTION(p, s(s(0)), s(s(0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0))), FUNCTION(p, s(s(s(x1))), s(s(s(x1))), s(x0))) by FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (131) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) by FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) We considered the (Usable) Rules:none And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) by FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (139) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [1] POL(p) = 0 POL(plus) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1))), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (145) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (149) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1^2 + [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [2] + x_1 ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) by FUNCTION(if, false, s(s(s(s(y0)))), z1) -> c8(FUNCTION(p, s(s(s(s(y0)))), s(s(s(s(y0)))), z1)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c7_1, c8_2, c8_1, c5_1, c4_1 ---------------------------------------- (153) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(x0)), z1) -> c7(FUNCTION(plus, z1, s(function(p, s(x0), x0, x0)), s(z1))) by FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(0), 0, 0)), s(s(x0)))) by FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (157) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1^2 + [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [2] + x_1 ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(x1))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(x1)), s(x1), s(x1))), s(s(x0)))) by FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(0)), s(0), s(0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(s(s(x1))), s(s(x1)), s(s(x1)))), s(s(x0)))) by FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (167) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) We considered the (Usable) Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) And the Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(FUNCTION(x_1, x_2, x_3, x_4)) = [2]x_1^2 + [2]x_1*x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(function(x_1, x_2, x_3, x_4)) = x_2 POL(if) = [2] POL(p) = 0 POL(plus) = [2] POL(s(x_1)) = [2] + x_1 ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(s(x0)), s(x0), s(x0)))), s(s(x1)))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_2, c8_1, c7_1, c5_1, c4_1 ---------------------------------------- (171) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(s(x0))), z1) -> c8(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1)), FUNCTION(p, s(s(s(x0))), s(s(s(x0))), z1)) by FUNCTION(if, false, s(s(s(z0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(z0), z0, z0))), s(s(x0))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0))), FUNCTION(p, s(s(s(0))), s(s(s(0))), s(x0))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0))), FUNCTION(p, s(s(s(s(x1)))), s(s(s(s(x1)))), s(x0))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (177) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (181) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (183) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(0)), z1) -> c7(FUNCTION(plus, z1, s(0), s(z1))) by FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c7_1, c8_2, c5_1, c4_1 ---------------------------------------- (185) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FUNCTION(if, false, s(s(s(x0))), z1) -> c7(FUNCTION(plus, z1, s(s(function(p, s(x0), x0, x0))), s(z1))) by FUNCTION(if, false, s(s(s(z0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(z0), z0, z0))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c5_1, c4_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(0), 0, 0))), s(s(x0)))) by FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c5_1, c4_1 ---------------------------------------- (189) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c5_1, c4_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FUNCTION(if, false, s(s(s(s(x1)))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(function(p, s(s(x1)), s(x1), s(x1)))), s(s(x0)))) by FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: function(p, s(0), z0, z1) -> 0 function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0)) Tuples: FUNCTION(if, false, s(s(s(x0))), x1) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), x1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(0))), z1) -> c7(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c8(FUNCTION(plus, z0, s(s(s(0))), s(z0)), FUNCTION(p, s(s(s(s(0)))), s(s(s(s(0)))), z0)) FUNCTION(if, false, s(s(0)), z1) -> c8(FUNCTION(plus, z1, s(0), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(0)))), z0) -> c7(FUNCTION(plus, z0, s(s(s(0))), s(z0))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(s(0))), z1) -> c8(FUNCTION(plus, z1, s(s(0)), s(z1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) S tuples: FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c8(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1)), FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), z1)) FUNCTION(if, false, s(s(s(s(x0)))), z1) -> c7(FUNCTION(plus, z1, s(s(s(function(p, s(x0), x0, x0)))), s(z1))) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c8(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1)), FUNCTION(p, s(s(s(s(s(z0))))), s(s(s(s(s(z0))))), z1)) FUNCTION(if, false, s(s(s(s(s(z0))))), z1) -> c7(FUNCTION(plus, z1, s(s(s(s(function(p, s(z0), z0, z0))))), s(z1))) FUNCTION(plus, x1, s(y0), s(x1)) -> c5(FUNCTION(if, false, s(y0), s(x1))) FUNCTION(plus, x1, s(s(y0)), s(x1)) -> c5(FUNCTION(if, false, s(s(y0)), s(x1))) FUNCTION(plus, x0, s(s(0)), s(x0)) -> c5(FUNCTION(if, false, s(s(0)), s(x0))) FUNCTION(plus, x1, s(s(s(y0))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(y0))), s(x1))) FUNCTION(plus, x0, s(s(s(0))), s(x0)) -> c5(FUNCTION(if, false, s(s(s(0))), s(x0))) FUNCTION(plus, x1, s(s(s(s(y0)))), s(x1)) -> c5(FUNCTION(if, false, s(s(s(s(y0)))), s(x1))) FUNCTION(p, s(s(s(s(y0)))), z1, z2) -> c4(FUNCTION(p, s(s(s(y0))), s(s(y0)), s(s(y0)))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0))), FUNCTION(p, s(s(z0)), s(s(z0)), s(x0))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1))), FUNCTION(p, s(s(s(z0))), s(s(s(z0))), s(x1))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c8(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1))), FUNCTION(p, s(s(s(s(z0)))), s(s(s(s(z0)))), s(x1))) FUNCTION(if, false, s(s(z0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(function(p, s(z0), z0, z0)), s(s(x0)))) FUNCTION(if, false, s(s(s(z0))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(function(p, s(z0), z0, z0))), s(s(x1)))) FUNCTION(if, false, s(s(s(s(z0)))), s(x1)) -> c7(FUNCTION(plus, s(x1), s(s(s(function(p, s(z0), z0, z0)))), s(s(x1)))) K tuples: FUNCTION(if, false, s(s(s(s(x0)))), x1) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), x1)) FUNCTION(if, false, s(s(s(x0))), s(x1)) -> c8(FUNCTION(p, s(s(s(x0))), s(s(s(x0))), s(x1))) FUNCTION(if, false, s(s(s(s(x0)))), s(x1)) -> c8(FUNCTION(p, s(s(s(s(x0)))), s(s(s(s(x0)))), s(x1))) FUNCTION(if, false, s(s(0)), s(x0)) -> c8(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c8(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) FUNCTION(if, false, s(s(0)), s(x0)) -> c7(FUNCTION(plus, s(x0), s(0), s(s(x0)))) FUNCTION(if, false, s(s(s(0))), s(x0)) -> c7(FUNCTION(plus, s(x0), s(s(0)), s(s(x0)))) Defined Rule Symbols: function_4 Defined Pair Symbols: FUNCTION_4 Compound Symbols: c8_1, c8_2, c7_1, c5_1, c4_1