WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 317 ms] (2) CpxRelTRS (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (4) CdtProblem (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 74 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms] (16) CdtProblem (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (18) BOUNDS(1, 1) (19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (20) TRS for Loop Detection (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (22) BEST (23) proven lower bound (24) LowerBoundPropagationProof [FINISHED, 0 ms] (25) BOUNDS(n^1, INF) (26) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 The (relative) TRS S consists of the following rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 The (relative) TRS S consists of the following rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 Tuples: MINUS'(0, z0) -> c10 MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(z0, 0) -> c12 GEQ'(0, s(z0)) -> c13 GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) DIV'(0, s(z0)) -> c15 DIV'(s(z0), s(z1)) -> c16(IF'(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) IF'(true, z0, z1) -> c17 IF'(false, z0, z1) -> c18 MINUS''(0, z0) -> c19 MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(z0, 0) -> c21 GEQ''(0, s(z0)) -> c22 GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(0, s(z0)) -> c24 DIV''(s(z0), s(z1)) -> c25(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) IF''(true, z0, z1) -> c27 IF''(false, z0, z1) -> c28 S tuples: MINUS''(0, z0) -> c19 MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(z0, 0) -> c21 GEQ''(0, s(z0)) -> c22 GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(0, s(z0)) -> c24 DIV''(s(z0), s(z1)) -> c25(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) IF''(true, z0, z1) -> c27 IF''(false, z0, z1) -> c28 K tuples:none Defined Rule Symbols: MINUS_2, GEQ_2, DIV_2, IF_3, minus_2, geq_2, div_2, if_3 Defined Pair Symbols: MINUS'_2, GEQ'_2, DIV'_2, IF'_3, MINUS''_2, GEQ''_2, DIV''_2, IF''_3 Compound Symbols: c10, c11_1, c12, c13, c14_1, c15, c16_4, c17, c18, c19, c20_1, c21, c22, c23_1, c24, c25_5, c26_7, c27, c28 ---------------------------------------- (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: GEQ''(z0, 0) -> c21 IF'(true, z0, z1) -> c17 DIV'(0, s(z0)) -> c15 GEQ'(z0, 0) -> c12 MINUS''(0, z0) -> c19 IF''(false, z0, z1) -> c28 GEQ''(0, s(z0)) -> c22 MINUS'(0, z0) -> c10 IF''(true, z0, z1) -> c27 IF'(false, z0, z1) -> c18 DIV''(0, s(z0)) -> c24 GEQ'(0, s(z0)) -> c13 ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) DIV'(s(z0), s(z1)) -> c16(IF'(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c25(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) S tuples: MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c25(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(IF''(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) K tuples:none Defined Rule Symbols: MINUS_2, GEQ_2, DIV_2, IF_3, minus_2, geq_2, div_2, if_3 Defined Pair Symbols: MINUS'_2, GEQ'_2, DIV'_2, MINUS''_2, GEQ''_2, DIV''_2 Compound Symbols: c11_1, c14_1, c16_4, c20_1, c23_1, c25_5, c26_7 ---------------------------------------- (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c25(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) S tuples: MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c25(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) K tuples:none Defined Rule Symbols: MINUS_2, GEQ_2, DIV_2, IF_3, minus_2, geq_2, div_2, if_3 Defined Pair Symbols: MINUS'_2, GEQ'_2, MINUS''_2, GEQ''_2, DIV'_2, DIV''_2 Compound Symbols: c11_1, c14_1, c20_1, c23_1, c16_3, c25_4, c26_6 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) S tuples: MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) K tuples:none Defined Rule Symbols: MINUS_2, GEQ_2, DIV_2, IF_3, minus_2, geq_2, div_2, if_3 Defined Pair Symbols: MINUS'_2, GEQ'_2, MINUS''_2, GEQ''_2, DIV'_2, DIV''_2 Compound Symbols: c11_1, c14_1, c20_1, c23_1, c16_3, c26_6, c10_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) S tuples: MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) K tuples:none Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS'_2, GEQ'_2, MINUS''_2, GEQ''_2, DIV'_2, DIV''_2 Compound Symbols: c11_1, c14_1, c20_1, c23_1, c16_3, c26_6, c10_1 ---------------------------------------- (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) We considered the (Usable) Rules: minus(s(z0), s(z1)) -> minus(z0, z1) minus(0, z0) -> 0 And the Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DIV'(x_1, x_2)) = x_1 POL(DIV''(x_1, x_2)) = [1] + x_1 + x_2 POL(GEQ'(x_1, x_2)) = x_1 POL(GEQ''(x_1, x_2)) = x_1 + x_2 POL(MINUS'(x_1, x_2)) = 0 POL(MINUS''(x_1, x_2)) = 0 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c20(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c26(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) S tuples: MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) K tuples: GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS'_2, GEQ'_2, MINUS''_2, GEQ''_2, DIV'_2, DIV''_2 Compound Symbols: c11_1, c14_1, c20_1, c23_1, c16_3, c26_6, c10_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) We considered the (Usable) Rules: minus(s(z0), s(z1)) -> minus(z0, z1) minus(0, z0) -> 0 And the Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DIV'(x_1, x_2)) = 0 POL(DIV''(x_1, x_2)) = x_1 POL(GEQ'(x_1, x_2)) = 0 POL(GEQ''(x_1, x_2)) = [1] POL(MINUS'(x_1, x_2)) = 0 POL(MINUS''(x_1, x_2)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c16(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c20(x_1)) = x_1 POL(c23(x_1)) = x_1 POL(c26(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 POL(minus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS'(s(z0), s(z1)) -> c11(MINUS'(z0, z1)) GEQ'(s(z0), s(z1)) -> c14(GEQ'(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV'(s(z0), s(z1)) -> c16(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) S tuples:none K tuples: GEQ''(s(z0), s(z1)) -> c23(GEQ''(z0, z1)) DIV''(s(z0), s(z1)) -> c26(GEQ'(z0, z1), DIV'(minus(z0, z1), s(z1)), MINUS'(z0, z1), DIV''(minus(z0, z1), s(z1)), MINUS'(z0, z1), MINUS''(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(DIV'(minus(z0, z1), s(z1))) DIV''(s(z0), s(z1)) -> c10(MINUS'(z0, z1)) DIV''(s(z0), s(z1)) -> c10(GEQ''(z0, z1)) MINUS''(s(z0), s(z1)) -> c20(MINUS''(z0, z1)) Defined Rule Symbols: minus_2 Defined Pair Symbols: MINUS'_2, GEQ'_2, MINUS''_2, GEQ''_2, DIV'_2, DIV''_2 Compound Symbols: c11_1, c14_1, c20_1, c23_1, c16_3, c26_6, c10_1 ---------------------------------------- (17) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (18) BOUNDS(1, 1) ---------------------------------------- (19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (20) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 The (relative) TRS S consists of the following rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Rewrite Strategy: INNERMOST ---------------------------------------- (21) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence MINUS(s(z0), s(z1)) ->^+ c1(MINUS(z0, z1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [z0 / s(z0), z1 / s(z1)]. The result substitution is [ ]. ---------------------------------------- (22) Complex Obligation (BEST) ---------------------------------------- (23) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 The (relative) TRS S consists of the following rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Rewrite Strategy: INNERMOST ---------------------------------------- (24) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (25) BOUNDS(n^1, INF) ---------------------------------------- (26) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: MINUS(0, z0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) GEQ(z0, 0) -> c2 GEQ(0, s(z0)) -> c3 GEQ(s(z0), s(z1)) -> c4(GEQ(z0, z1)) DIV(0, s(z0)) -> c5 DIV(s(z0), s(z1)) -> c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1)) DIV(s(z0), s(z1)) -> c7(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1)) IF(true, z0, z1) -> c8 IF(false, z0, z1) -> c9 The (relative) TRS S consists of the following rules: minus(0, z0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) geq(z0, 0) -> true geq(0, s(z0)) -> false geq(s(z0), s(z1)) -> geq(z0, z1) div(0, s(z0)) -> 0 div(s(z0), s(z1)) -> if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 Rewrite Strategy: INNERMOST