KILLED proof of input_ajhKP9HHZm.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 359 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 144 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 480 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 168 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 1328 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 258 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 [ComplexityIfPolyImplication, 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (74) CdtProblem (75) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (78) CdtProblem (79) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [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), 7 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) 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: times(x, y) -> help(x, y, 0) help(x, y, c) -> if(lt(c, y), x, y, c) if(true, x, y, c) -> plus(x, help(x, y, s(c))) if(false, x, y, c) -> 0 lt(0, s(x)) -> true lt(s(x), 0) -> false lt(s(x), s(y)) -> lt(x, y) plus(x, 0) -> x plus(0, x) -> x plus(x, s(y)) -> s(plus(x, y)) plus(s(x), y) -> s(plus(x, y)) 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: times(x, y) -> help(x, y, 0') help(x, y, c) -> if(lt(c, y), x, y, c) if(true, x, y, c) -> plus(x, help(x, y, s(c))) if(false, x, y, c) -> 0' lt(0', s(x)) -> true lt(s(x), 0') -> false lt(s(x), s(y)) -> lt(x, y) plus(x, 0') -> x plus(0', x) -> x plus(x, s(y)) -> s(plus(x, y)) plus(s(x), y) -> s(plus(x, y)) 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: times(x, y) -> help(x, y, 0) help(x, y, c) -> if(lt(c, y), x, y, c) if(true, x, y, c) -> plus(x, help(x, y, s(c))) if(false, x, y, c) -> 0 lt(0, s(x)) -> true lt(s(x), 0) -> false lt(s(x), s(y)) -> lt(x, y) plus(x, 0) -> x plus(0, x) -> x plus(x, s(y)) -> s(plus(x, y)) plus(s(x), y) -> s(plus(x, y)) 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: times(x, y) -> help(x, y, 0) [1] help(x, y, c) -> if(lt(c, y), x, y, c) [1] if(true, x, y, c) -> plus(x, help(x, y, s(c))) [1] if(false, x, y, c) -> 0 [1] lt(0, s(x)) -> true [1] lt(s(x), 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] plus(s(x), y) -> s(plus(x, y)) [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: times(x, y) -> help(x, y, 0) [1] help(x, y, c) -> if(lt(c, y), x, y, c) [1] if(true, x, y, c) -> plus(x, help(x, y, s(c))) [1] if(false, x, y, c) -> 0 [1] lt(0, s(x)) -> true [1] lt(s(x), 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] plus(s(x), y) -> s(plus(x, y)) [1] The TRS has the following type information: times :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false true :: true:false plus :: 0:s -> 0:s -> 0:s s :: 0:s -> 0:s false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: times_2 (c) The following functions are completely defined: lt_2 help_3 if_4 plus_2 Due to the following rules being added: lt(v0, v1) -> null_lt [0] if(v0, v1, v2, v3) -> 0 [0] And the following fresh constants: null_lt ---------------------------------------- (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: times(x, y) -> help(x, y, 0) [1] help(x, y, c) -> if(lt(c, y), x, y, c) [1] if(true, x, y, c) -> plus(x, help(x, y, s(c))) [1] if(false, x, y, c) -> 0 [1] lt(0, s(x)) -> true [1] lt(s(x), 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(v0, v1) -> null_lt [0] if(v0, v1, v2, v3) -> 0 [0] The TRS has the following type information: times :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s if :: true:false:null_lt -> 0:s -> 0:s -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false:null_lt true :: true:false:null_lt plus :: 0:s -> 0:s -> 0:s s :: 0:s -> 0:s false :: true:false:null_lt null_lt :: true:false:null_lt 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: times(x, y) -> help(x, y, 0) [1] help(x, s(x'), 0) -> if(true, x, s(x'), 0) [2] help(x, 0, s(x'')) -> if(false, x, 0, s(x'')) [2] help(x, s(y'), s(x1)) -> if(lt(x1, y'), x, s(y'), s(x1)) [2] help(x, y, c) -> if(null_lt, x, y, c) [1] if(true, x, y, c) -> plus(x, if(lt(s(c), y), x, y, s(c))) [2] if(false, x, y, c) -> 0 [1] lt(0, s(x)) -> true [1] lt(s(x), 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(v0, v1) -> null_lt [0] if(v0, v1, v2, v3) -> 0 [0] The TRS has the following type information: times :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s 0 :: 0:s if :: true:false:null_lt -> 0:s -> 0:s -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false:null_lt true :: true:false:null_lt plus :: 0:s -> 0:s -> 0:s s :: 0:s -> 0:s false :: true:false:null_lt null_lt :: true:false:null_lt Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 null_lt => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(x1, y'), x, 1 + y', 1 + x1) :|: x1 >= 0, z'' = 1 + x1, x >= 0, y' >= 0, z' = 1 + y', z = x help(z, z', z'') -{ 2 }-> if(2, x, 1 + x', 0) :|: z'' = 0, z' = 1 + x', x >= 0, x' >= 0, z = x help(z, z', z'') -{ 2 }-> if(1, x, 0, 1 + x'') :|: x >= 0, z'' = 1 + x'', x'' >= 0, z = x, z' = 0 help(z, z', z'') -{ 1 }-> if(0, x, y, c) :|: c >= 0, x >= 0, y >= 0, z = x, z' = y, z'' = c if(z, z', z'', z1) -{ 2 }-> plus(x, if(lt(1 + c, y), x, y, 1 + c)) :|: z = 2, z' = x, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + 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') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> help(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y ---------------------------------------- (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), z, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 2 }-> plus(z', if(lt(1 + z1, z''), z', z'', 1 + z1)) :|: z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lt } { plus } { if } { help } { times } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z' - 1), z, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 2 }-> plus(z', if(lt(1 + z1, z''), z', z'', 1 + z1)) :|: z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {plus}, {if}, {help}, {times} ---------------------------------------- (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), z, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 2 }-> plus(z', if(lt(1 + z1, z''), z', z'', 1 + z1)) :|: z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {plus}, {if}, {help}, {times} ---------------------------------------- (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: 2 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(lt(z'' - 1, z' - 1), z, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 2 }-> plus(z', if(lt(1 + z1, z''), z', z'', 1 + z1)) :|: z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {plus}, {if}, {help}, {times} Previous analysis results are: lt: runtime: ?, size: O(1) [2] ---------------------------------------- (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), z, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 2 }-> plus(z', if(lt(1 + z1, z''), z', z'', 1 + z1)) :|: z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {plus}, {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] ---------------------------------------- (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, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {plus}, {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] ---------------------------------------- (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, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {plus}, {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] 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 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z' }-> if(s, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: 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 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] plus: runtime: O(n^1) [1 + z + 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, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z + (z' - 1), z >= 0, z' - 1 >= 0 plus(z, z') -{ 1 + z + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) 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, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z + (z' - 1), z >= 0, z' - 1 >= 0 plus(z, z') -{ 1 + z + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z'] if: runtime: ?, size: INF ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if 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, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0 help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0 if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z + (z' - 1), z >= 0, z' - 1 >= 0 plus(z, z') -{ 1 + z + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {help}, {times} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [2] plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z'] if: runtime: INF, 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: lt(v0, v1) -> null_lt [0] if(v0, v1, v2, v3) -> null_if [0] plus(v0, v1) -> null_plus [0] And the following fresh constants: null_lt, null_if, 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: times(x, y) -> help(x, y, 0) [1] help(x, y, c) -> if(lt(c, y), x, y, c) [1] if(true, x, y, c) -> plus(x, help(x, y, s(c))) [1] if(false, x, y, c) -> 0 [1] lt(0, s(x)) -> true [1] lt(s(x), 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(v0, v1) -> null_lt [0] if(v0, v1, v2, v3) -> null_if [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: times :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus help :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus 0 :: 0:s:null_if:null_plus if :: true:false:null_lt -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus lt :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> true:false:null_lt true :: true:false:null_lt plus :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus s :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus false :: true:false:null_lt null_lt :: true:false:null_lt null_if :: 0:s:null_if:null_plus null_plus :: 0:s: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 true => 2 false => 1 null_lt => 0 null_if => 0 null_plus => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 1 }-> if(lt(c, y), x, y, c) :|: c >= 0, x >= 0, y >= 0, z = x, z' = y, z'' = c if(z, z', z'', z1) -{ 1 }-> plus(x, help(x, y, 1 + c)) :|: z = 2, z' = x, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0 if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0 if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0 lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + 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') -{ 1 }-> x :|: z' = x, x >= 0, 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 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> help(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y 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: times(z0, z1) -> help(z0, z1, 0) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: TIMES(z0, z1) -> c(HELP(z0, z1, 0)) HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) IF(false, z0, z1, z2) -> c3 LT(0, s(z0)) -> c4 LT(s(z0), 0) -> c5 LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, 0) -> c7 PLUS(0, z0) -> c8 PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) S tuples: TIMES(z0, z1) -> c(HELP(z0, z1, 0)) HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) IF(false, z0, z1, z2) -> c3 LT(0, s(z0)) -> c4 LT(s(z0), 0) -> c5 LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, 0) -> c7 PLUS(0, z0) -> c8 PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) K tuples:none Defined Rule Symbols: times_2, help_3, if_4, lt_2, plus_2 Defined Pair Symbols: TIMES_2, HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c_1, c1_2, c2_2, c3, c4, c5, c6_1, c7, c8, c9_1, c10_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: TIMES(z0, z1) -> c(HELP(z0, z1, 0)) Removed 5 trailing nodes: PLUS(z0, 0) -> c7 PLUS(0, z0) -> c8 LT(0, s(z0)) -> c4 LT(s(z0), 0) -> c5 IF(false, z0, z1, z2) -> c3 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: times(z0, z1) -> help(z0, z1, 0) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) S tuples: HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) K tuples:none Defined Rule Symbols: times_2, help_3, if_4, lt_2, plus_2 Defined Pair Symbols: HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c1_2, c2_2, c6_1, c9_1, c10_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: times(z0, z1) -> help(z0, z1, 0) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) S tuples: HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) K tuples:none Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: HELP_3, IF_4, LT_2, PLUS_2 Compound Symbols: c1_2, c2_2, c6_1, c9_1, c10_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) by HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0))) HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0))) HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) S tuples: IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0))) HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) K tuples:none Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, LT_2, PLUS_2, HELP_3 Compound Symbols: c2_2, c6_1, c9_1, c10_1, c1_2 ---------------------------------------- (49) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0))) Removed 1 trailing nodes: HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) S tuples: IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) K tuples:none Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, LT_2, PLUS_2, HELP_3 Compound Symbols: c2_2, c6_1, c9_1, c10_1, c1_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) by IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) K tuples:none Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: LT_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c6_1, c9_1, c10_1, c1_2, c2_2, c2_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) by HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(0), s(s(z0))) -> c1(IF(false, x0, s(0), s(s(z0))), LT(s(s(z0)), s(0))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(0), s(s(z0))) -> c1(IF(false, x0, s(0), s(s(z0))), LT(s(s(z0)), s(0))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(0), s(s(z0))) -> c1(IF(false, x0, s(0), s(s(z0))), LT(s(s(z0)), s(0))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) K tuples:none Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c6_1, c9_1, c10_1, c2_2, c2_1, c1_2, c1_1 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) K tuples:none Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c6_1, c9_1, c10_1, c2_2, c2_1, c1_2, c1_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) We considered the (Usable) Rules: lt(s(z0), 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) And the Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(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] + x_2 + x_3 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4 POL(LT(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(false) = [1] POL(help(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(lt(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: LT_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c6_1, c9_1, c10_1, c2_2, c2_1, c1_2, c1_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(z0), s(z1)) -> c6(LT(z0, z1)) by LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, IF_4, HELP_3, LT_2 Compound Symbols: c9_1, c10_1, c2_2, c2_1, c1_2, c1_1, c6_1 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0)))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, IF_4, HELP_3, LT_2 Compound Symbols: c9_1, c10_1, c2_2, c2_1, c1_2, c1_1, c6_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, IF_4, HELP_3, LT_2 Compound Symbols: c9_1, c10_1, c2_2, c2_1, c1_2, c1_1, c6_1 ---------------------------------------- (65) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) by IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, IF_4, HELP_3, LT_2 Compound Symbols: c9_1, c10_1, c2_1, c1_2, c1_1, c6_1, c2_2 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) by IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, IF_4, HELP_3, LT_2 Compound Symbols: c9_1, c10_1, c2_1, c1_2, c1_1, c6_1, c2_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) by IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, IF_4, HELP_3, LT_2 Compound Symbols: c9_1, c10_1, c2_1, c1_2, c1_1, c6_1, c2_2 ---------------------------------------- (71) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) by IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1 ---------------------------------------- (73) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0))) IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1 ---------------------------------------- (75) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) IF(true, x0, s(s(x1)), s(0)) -> c(HELP(x0, s(s(x1)), s(s(0)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) IF(true, x0, s(s(x1)), s(0)) -> c(HELP(x0, s(s(x1)), s(s(0)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1, c_1 ---------------------------------------- (77) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF(true, x0, s(s(x1)), s(0)) -> c(HELP(x0, s(s(x1)), s(s(0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1, c_1 ---------------------------------------- (79) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1, c_1 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) by IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_2, c1_1, c6_1, c2_1, c_1, c2_2 ---------------------------------------- (83) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) by HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) K tuples: HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, HELP_3, LT_2, IF_4 Compound Symbols: c9_1, c10_1, c1_1, c6_1, c2_1, c_1, c2_2, c1_2 ---------------------------------------- (85) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) by HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_4, HELP_3 Compound Symbols: c9_1, c10_1, c6_1, c2_1, c_1, c2_2, c1_2, c1_1 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) by IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_4, HELP_3 Compound Symbols: c9_1, c10_1, c6_1, c2_1, c2_2, c1_2, c1_1, c_1 ---------------------------------------- (89) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) by IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_4, HELP_3 Compound Symbols: c9_1, c10_1, c6_1, c2_1, c2_2, c1_2, c1_1, c_1 ---------------------------------------- (91) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) by IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_4, HELP_3 Compound Symbols: c9_1, c10_1, c6_1, c2_2, c1_2, c1_1, c_1, c2_1 ---------------------------------------- (93) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) by IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, HELP_3, IF_4 Compound Symbols: c9_1, c10_1, c6_1, c1_2, c1_1, c_1, c2_1, c2_2 ---------------------------------------- (95) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) by HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, HELP_3, IF_4 Compound Symbols: c9_1, c10_1, c6_1, c1_1, c_1, c2_1, c2_2, c1_2 ---------------------------------------- (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) by HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) S tuples: PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_4, HELP_3 Compound Symbols: c9_1, c10_1, c6_1, c_1, c2_1, c2_2, c1_2, c1_1 ---------------------------------------- (99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) by PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) S tuples: PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_4, HELP_3 Compound Symbols: c10_1, c6_1, c_1, c2_1, c2_2, c1_2, c1_1, c9_1 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) by PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) S tuples: LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: LT_2, IF_4, HELP_3, PLUS_2 Compound Symbols: c6_1, c_1, c2_1, c2_2, c1_2, c1_1, c9_1, c10_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) by LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c1_1, c9_1, c10_1, c6_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) by HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c9_1, c10_1, c6_1, c1_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) by PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c9_1, c10_1, c6_1, c1_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) by PLUS(s(z0), s(s(y1))) -> c9(PLUS(s(z0), s(y1))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(y1)))) -> c9(PLUS(s(z0), s(s(y1)))) PLUS(s(s(y0)), s(s(y1))) -> c9(PLUS(s(s(y0)), s(y1))) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c10_1, c6_1, c1_1, c9_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) by PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c10_1, c6_1, c1_1, c9_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) by PLUS(s(s(y0)), s(s(z1))) -> c10(PLUS(s(y0), s(s(z1)))) PLUS(s(s(s(y0))), s(s(z1))) -> c10(PLUS(s(s(y0)), s(s(z1)))) PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c10(PLUS(s(y0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(s(y1))))) -> c10(PLUS(s(y0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1))))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c10_1, c6_1, c1_1, c9_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) by PLUS(s(s(s(y0))), s(z1)) -> c10(PLUS(s(s(y0)), s(z1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(z1)) -> c10(PLUS(s(s(s(y0))), s(z1))) PLUS(s(s(s(s(y0)))), s(s(y1))) -> c10(PLUS(s(s(s(y0))), s(s(y1)))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(s(s(s(y0))))), s(s(y1))) -> c10(PLUS(s(s(s(s(y0)))), s(s(y1)))) PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c10(PLUS(s(s(s(y0))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c10(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(s(z0), 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2) if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2))) if(false, z0, z1, z2) -> 0 plus(z0, 0) -> z0 plus(0, z0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(z1)) -> c10(PLUS(s(s(s(y0))), s(z1))) PLUS(s(s(s(s(s(y0))))), s(s(y1))) -> c10(PLUS(s(s(s(s(y0)))), s(s(y1)))) PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c10(PLUS(s(s(s(y0))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c10(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) S tuples: IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2)))))) IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2)))))) HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1)))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1)))) PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1))) PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1))) PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1)))) PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1)) PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1)) PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1)))) PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1))) PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1))))) PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1)))) PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1)))))) PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1))))) PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1)))) PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1)))))) PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(s(s(y0)))), s(z1)) -> c10(PLUS(s(s(s(y0))), s(z1))) PLUS(s(s(s(s(s(y0))))), s(s(y1))) -> c10(PLUS(s(s(s(s(y0)))), s(s(y1)))) PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c10(PLUS(s(s(s(y0))), s(s(s(s(y1)))))) PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c10(PLUS(s(s(s(s(y0)))), s(s(s(y1))))) K tuples: IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, help_3, if_4, plus_2 Defined Pair Symbols: IF_4, HELP_3, LT_2, PLUS_2 Compound Symbols: c_1, c2_1, c2_2, c1_2, c6_1, c1_1, c9_1, c10_1