KILLED proof of input_OVo0kenmXT.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 3 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 336 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 172 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 267 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 9545 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 6860 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 657 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms] (66) CdtProblem (67) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (70) CdtProblem (71) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (94) CdtProblem (95) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) 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: lt(x, 0) -> false lt(0, s(y)) -> true lt(s(x), s(y)) -> lt(x, y) plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) quot(x, s(y)) -> help(x, s(y), 0) help(x, s(y), c) -> if(lt(c, x), x, s(y), c) if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) if(false, x, s(y), c) -> 0 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: lt(x, 0') -> false lt(0', s(y)) -> true lt(s(x), s(y)) -> lt(x, y) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) quot(x, s(y)) -> help(x, s(y), 0') help(x, s(y), c) -> if(lt(c, x), x, s(y), c) if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) if(false, x, s(y), c) -> 0' 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: lt(x, 0) -> false lt(0, s(y)) -> true lt(s(x), s(y)) -> lt(x, y) plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) quot(x, s(y)) -> help(x, s(y), 0) help(x, s(y), c) -> if(lt(c, x), x, s(y), c) if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) if(false, x, s(y), c) -> 0 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: lt(x, 0) -> false [1] lt(0, s(y)) -> true [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] quot(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1] if(false, x, s(y), c) -> 0 [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: lt(x, 0) -> false [1] lt(0, s(y)) -> true [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] quot(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1] if(false, x, s(y), c) -> 0 [1] The TRS has the following type information: lt :: 0:s -> 0:s -> false:true 0 :: 0:s false :: false:true s :: 0:s -> 0:s true :: false:true plus :: 0:s -> 0:s -> 0:s quot :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s if :: false:true -> 0:s -> 0:s -> 0:s -> 0:s 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: quot_2 help_3 if_4 (c) The following functions are completely defined: lt_2 plus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt(x, 0) -> false [1] lt(0, s(y)) -> true [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] quot(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1] if(false, x, s(y), c) -> 0 [1] The TRS has the following type information: lt :: 0:s -> 0:s -> false:true 0 :: 0:s false :: false:true s :: 0:s -> 0:s true :: false:true plus :: 0:s -> 0:s -> 0:s quot :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s if :: false:true -> 0:s -> 0:s -> 0:s -> 0:s 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: lt(x, 0) -> false [1] lt(0, s(y)) -> true [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] quot(x, s(y)) -> help(x, s(y), 0) [1] help(0, s(y), c) -> if(false, 0, s(y), c) [2] help(s(y'), s(y), 0) -> if(true, s(y'), s(y), 0) [2] help(s(y''), s(y), s(x')) -> if(lt(x', y''), s(y''), s(y), s(x')) [2] if(true, x, s(y), c) -> s(help(x, s(y), s(plus(c, y)))) [2] if(false, x, s(y), c) -> 0 [1] The TRS has the following type information: lt :: 0:s -> 0:s -> false:true 0 :: 0:s false :: false:true s :: 0:s -> 0:s true :: false:true plus :: 0:s -> 0:s -> 0:s quot :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s if :: false:true -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 0 true => 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(x', y''), 1 + y'', 1 + y, 1 + x') :|: z' = 1 + y, z = 1 + y'', z'' = 1 + x', y >= 0, x' >= 0, y'' >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + y', 1 + y, 0) :|: z' = 1 + y, z'' = 0, y >= 0, z = 1 + y', y' >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + y, c) :|: z' = 1 + y, c >= 0, y >= 0, z = 0, z'' = c if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z1 = c, c >= 0, x >= 0, y >= 0, z'' = 1 + y, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(x, 1 + y, 1 + plus(c, y)) :|: z' = x, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z'' = 1 + y lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x quot(z, z') -{ 1 }-> help(x, 1 + y, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x ---------------------------------------- (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: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lt } { plus } { help, if } { quot } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {lt}, {plus}, {help,if}, {quot} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {lt}, {plus}, {help,if}, {quot} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {lt}, {plus}, {help,if}, {quot} Previous analysis results are: lt: runtime: ?, size: O(1) [1] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {plus}, {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {plus}, {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {plus}, {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 + z'' }-> 1 + help(z', 1 + (z'' - 1), 1 + s1) :|: s1 >= 0, s1 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: help after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 + z'' }-> 1 + help(z', 1 + (z'' - 1), 1 + s1) :|: s1 >= 0, s1 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] help: runtime: ?, size: INF if: runtime: ?, size: INF ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: help after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 + z'' }-> 1 + help(z', 1 + (z'' - 1), 1 + s1) :|: s1 >= 0, s1 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0 quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {quot} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] help: runtime: INF, size: INF if: runtime: ?, size: INF ---------------------------------------- (37) 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: quot(v0, v1) -> null_quot [0] help(v0, v1, v2) -> null_help [0] if(v0, v1, v2, v3) -> null_if [0] lt(v0, v1) -> null_lt [0] plus(v0, v1) -> null_plus [0] And the following fresh constants: null_quot, null_help, null_if, null_lt, null_plus ---------------------------------------- (38) 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: lt(x, 0) -> false [1] lt(0, s(y)) -> true [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] quot(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1] if(false, x, s(y), c) -> 0 [1] quot(v0, v1) -> null_quot [0] help(v0, v1, v2) -> null_help [0] if(v0, v1, v2, v3) -> null_if [0] lt(v0, v1) -> null_lt [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: lt :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> false:true:null_lt 0 :: 0:s:null_quot:null_help:null_if:null_plus false :: false:true:null_lt s :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus true :: false:true:null_lt plus :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus quot :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus help :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus if :: false:true:null_lt -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus null_quot :: 0:s:null_quot:null_help:null_if:null_plus null_help :: 0:s:null_quot:null_help:null_if:null_plus null_if :: 0:s:null_quot:null_help:null_if:null_plus null_lt :: false:true:null_lt null_plus :: 0:s:null_quot:null_help:null_if:null_plus Rewrite Strategy: INNERMOST ---------------------------------------- (39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 1 true => 2 null_quot => 0 null_help => 0 null_if => 0 null_lt => 0 null_plus => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 1 }-> if(lt(c, x), x, 1 + y, c) :|: z' = 1 + y, c >= 0, x >= 0, y >= 0, z = x, z'' = c help(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z'' = 1 + y if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + help(x, 1 + y, plus(c, 1 + y)) :|: z = 2, z' = x, z1 = c, c >= 0, x >= 0, y >= 0, z'' = 1 + y lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x quot(z, z') -{ 1 }-> help(x, 1 + y, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) quot(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1)))) if(false, z0, s(z1), z2) -> 0 Tuples: LT(z0, 0) -> c LT(0, s(z0)) -> c1 LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, 0) -> c3 PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0)) HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c8 S tuples: LT(z0, 0) -> c LT(0, s(z0)) -> c1 LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, 0) -> c3 PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0)) HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c8 K tuples:none Defined Rule Symbols: lt_2, plus_2, quot_2, help_3, if_4 Defined Pair Symbols: LT_2, PLUS_2, QUOT_2, HELP_3, IF_4 Compound Symbols: c, c1, c2_1, c3, c4_1, c5_1, c6_2, c7_2, c8 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0)) Removed 4 trailing nodes: IF(false, z0, s(z1), z2) -> c8 LT(0, s(z0)) -> c1 PLUS(z0, 0) -> c3 LT(z0, 0) -> c ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) quot(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1)))) if(false, z0, s(z1), z2) -> 0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2, quot_2, help_3, if_4 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c7_2 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: quot(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1)))) if(false, z0, s(z1), z2) -> 0 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c7_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) by HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0)) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0)) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0)) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2 ---------------------------------------- (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2, c6_1 ---------------------------------------- (53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) We considered the (Usable) Rules: plus(z0, s(z1)) -> s(plus(z0, z1)) And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(HELP(x_1, x_2, x_3)) = [3]x_2 + [2]x_3 POL(IF(x_1, x_2, x_3, x_4)) = [3]x_3 POL(LT(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2)) = x_1 + x_2 POL(false) = [3] POL(lt(x_1, x_2)) = [3] + [3]x_1 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(true) = [3] ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) K tuples: HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2, c6_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) by IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) K tuples: HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c6_1, c7_2 ---------------------------------------- (57) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c7_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) by HELP(s(0), s(x1), s(z0)) -> c6(IF(false, s(0), s(x1), s(z0)), LT(s(z0), s(0))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(0), s(x1), s(z0)) -> c6(IF(false, s(0), s(x1), s(z0)), LT(s(z0), s(0))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(0), s(x1), s(z0)) -> c6(IF(false, s(0), s(x1), s(z0)), LT(s(z0), s(0))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2, c6_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2, c6_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) We considered the (Usable) Rules:none And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(HELP(x_1, x_2, x_3)) = x_1 POL(IF(x_1, x_2, x_3, x_4)) = x_2 POL(LT(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2)) = x_1 + x_2 POL(false) = [1] POL(lt(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = x_1^3 + x_1^2*x_2 + x_1*x_2^2 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) K tuples: HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2, c6_1 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) We considered the (Usable) Rules:none And the Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(HELP(x_1, x_2, x_3)) = [1] POL(IF(x_1, x_2, x_3, x_4)) = [1] POL(LT(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2)) = x_1 + x_2 POL(false) = [3] POL(lt(x_1, x_2)) = [3] + [3]x_1 POL(plus(x_1, x_2)) = [3]x_1 + [2]x_2 POL(s(x_1)) = [2] POL(true) = [3] ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) K tuples: HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c4_1, c7_2, c6_2, c6_1 ---------------------------------------- (67) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) by IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) K tuples: HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c6_1, c7_2 ---------------------------------------- (69) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c6_1, c7_2 ---------------------------------------- (71) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) by HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c2(LT(z0, z1)) PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) K tuples: HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c4_1, c6_2, c7_2, c6_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(z0), s(z1)) -> c2(LT(z0, z1)) by LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) S tuples: PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0)))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) K tuples: HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, IF_4, LT_2 Compound Symbols: c4_1, c6_2, c7_2, c6_1, c2_1 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) S tuples: PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) K tuples: HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, IF_4, LT_2 Compound Symbols: c4_1, c6_2, c7_2, c6_1, c2_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) by PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) K tuples: HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c6_2, c7_2, c6_1, c2_1, c4_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) by HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c6_2, c7_2, c2_1, c6_1, c4_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) by IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))), PLUS(s(0), s(0))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))), PLUS(s(0), s(0))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))), PLUS(s(0), s(0))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c6_2, c7_2, c2_1, c6_1, c4_1 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0)))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c6_2, c7_2, c2_1, c6_1, c4_1, c7_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) by IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, LT_2, PLUS_2, IF_4 Compound Symbols: c6_2, c2_1, c6_1, c4_1, c7_2, c7_1 ---------------------------------------- (87) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0))) IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0)))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, LT_2, PLUS_2, IF_4 Compound Symbols: c6_2, c2_1, c4_1, c6_1, c7_2 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, LT_2, PLUS_2, IF_4 Compound Symbols: c6_2, c2_1, c4_1, c6_1, c7_2, c7_1 ---------------------------------------- (91) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(PLUS(s(0), s(s(z1)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(PLUS(s(0), s(s(z1)))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, LT_2, PLUS_2, IF_4 Compound Symbols: c6_2, c2_1, c4_1, c6_1, c7_2, c7_1, c_1 ---------------------------------------- (93) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(PLUS(s(0), s(s(z1)))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, LT_2, PLUS_2, IF_4 Compound Symbols: c6_2, c2_1, c4_1, c6_1, c7_2, c7_1, c_1 ---------------------------------------- (95) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, LT_2, PLUS_2, IF_4 Compound Symbols: c6_2, c2_1, c4_1, c6_1, c7_2, c7_1, c_1 ---------------------------------------- (97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1))) by LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: HELP_3, PLUS_2, IF_4, LT_2 Compound Symbols: c6_2, c4_1, c6_1, c7_2, c7_1, c_1, c2_1 ---------------------------------------- (99) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) by HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) S tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) K tuples: HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, IF_4, LT_2 Compound Symbols: c4_1, c6_1, c7_2, c7_1, c_1, c2_1, c6_2 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) by HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) S tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, IF_4, LT_2, HELP_3 Compound Symbols: c4_1, c7_2, c7_1, c_1, c2_1, c6_2, c6_1 ---------------------------------------- (103) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) by IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) S tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) K tuples: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, IF_4, LT_2, HELP_3 Compound Symbols: c4_1, c7_2, c_1, c2_1, c6_2, c6_1, c7_1 ---------------------------------------- (105) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) by HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) S tuples: PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, IF_4, LT_2, HELP_3 Compound Symbols: c4_1, c7_2, c_1, c2_1, c6_2, c6_1, c7_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) by PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_4, LT_2, HELP_3, PLUS_2 Compound Symbols: c7_2, c_1, c2_1, c6_2, c6_1, c7_1, c4_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) by HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_4, LT_2, HELP_3, PLUS_2 Compound Symbols: c7_2, c_1, c2_1, c6_2, c6_1, c7_1, c4_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) by HELP(s(s(s(y1))), s(0), s(s(s(z1)))) -> c6(LT(s(s(s(z1))), s(s(s(y1))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 Tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(0), s(s(s(z1)))) -> c6(LT(s(s(s(z1))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(0), s(s(s(z1)))) -> c6(LT(s(s(s(z1))), s(s(s(y1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_4, LT_2, HELP_3, PLUS_2 Compound Symbols: c7_2, c_1, c2_1, c6_2, c7_1, c4_1, c6_1