/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1011 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(s(N)) -> N +(N, 0) -> N +(s(N), s(M)) -> s(s(+(N, M))) *(N, 0) -> 0 *(s(N), s(M)) -> s(+(N, +(M, *(N, M)))) gt(0, M) -> False gt(NzN, 0) -> u_4(is_NzNat(NzN)) u_4(True) -> True is_NzNat(0) -> False is_NzNat(s(N)) -> True gt(s(N), s(M)) -> gt(N, M) lt(N, M) -> gt(M, N) d(0, N) -> N d(s(N), s(M)) -> d(N, M) quot(N, NzM) -> u_11(is_NzNat(NzM), N, NzM) u_11(True, N, NzM) -> u_1(gt(N, NzM), N, NzM) u_1(True, N, NzM) -> s(quot(d(N, NzM), NzM)) quot(NzM, NzM) -> u_01(is_NzNat(NzM)) u_01(True) -> s(0) quot(N, NzM) -> u_21(is_NzNat(NzM), NzM, N) u_21(True, NzM, N) -> u_2(gt(NzM, N)) u_2(True) -> 0 gcd(0, N) -> 0 gcd(NzM, NzM) -> u_02(is_NzNat(NzM), NzM) u_02(True, NzM) -> NzM gcd(NzN, NzM) -> u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM) u_31(True, True, NzN, NzM) -> u_3(gt(NzN, NzM), NzN, NzM) u_3(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(False) -> False encArg(True) -> True encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_u_4(x_1)) -> u_4(encArg(x_1)) encArg(cons_is_NzNat(x_1)) -> is_NzNat(encArg(x_1)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_u_11(x_1, x_2, x_3)) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_1(x_1, x_2, x_3)) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_01(x_1)) -> u_01(encArg(x_1)) encArg(cons_u_21(x_1, x_2, x_3)) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_2(x_1)) -> u_2(encArg(x_1)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_u_02(x_1, x_2)) -> u_02(encArg(x_1), encArg(x_2)) encArg(cons_u_31(x_1, x_2, x_3, x_4)) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u_3(x_1, x_2, x_3)) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_False -> False encode_u_4(x_1) -> u_4(encArg(x_1)) encode_is_NzNat(x_1) -> is_NzNat(encArg(x_1)) encode_True -> True encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_u_11(x_1, x_2, x_3) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_1(x_1, x_2, x_3) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_01(x_1) -> u_01(encArg(x_1)) encode_u_21(x_1, x_2, x_3) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_2(x_1) -> u_2(encArg(x_1)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_u_02(x_1, x_2) -> u_02(encArg(x_1), encArg(x_2)) encode_u_31(x_1, x_2, x_3, x_4) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u_3(x_1, x_2, x_3) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(s(N)) -> N +(N, 0) -> N +(s(N), s(M)) -> s(s(+(N, M))) *(N, 0) -> 0 *(s(N), s(M)) -> s(+(N, +(M, *(N, M)))) gt(0, M) -> False gt(NzN, 0) -> u_4(is_NzNat(NzN)) u_4(True) -> True is_NzNat(0) -> False is_NzNat(s(N)) -> True gt(s(N), s(M)) -> gt(N, M) lt(N, M) -> gt(M, N) d(0, N) -> N d(s(N), s(M)) -> d(N, M) quot(N, NzM) -> u_11(is_NzNat(NzM), N, NzM) u_11(True, N, NzM) -> u_1(gt(N, NzM), N, NzM) u_1(True, N, NzM) -> s(quot(d(N, NzM), NzM)) quot(NzM, NzM) -> u_01(is_NzNat(NzM)) u_01(True) -> s(0) quot(N, NzM) -> u_21(is_NzNat(NzM), NzM, N) u_21(True, NzM, N) -> u_2(gt(NzM, N)) u_2(True) -> 0 gcd(0, N) -> 0 gcd(NzM, NzM) -> u_02(is_NzNat(NzM), NzM) u_02(True, NzM) -> NzM gcd(NzN, NzM) -> u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM) u_31(True, True, NzN, NzM) -> u_3(gt(NzN, NzM), NzN, NzM) u_3(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(False) -> False encArg(True) -> True encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_u_4(x_1)) -> u_4(encArg(x_1)) encArg(cons_is_NzNat(x_1)) -> is_NzNat(encArg(x_1)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_u_11(x_1, x_2, x_3)) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_1(x_1, x_2, x_3)) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_01(x_1)) -> u_01(encArg(x_1)) encArg(cons_u_21(x_1, x_2, x_3)) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_2(x_1)) -> u_2(encArg(x_1)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_u_02(x_1, x_2)) -> u_02(encArg(x_1), encArg(x_2)) encArg(cons_u_31(x_1, x_2, x_3, x_4)) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u_3(x_1, x_2, x_3)) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_False -> False encode_u_4(x_1) -> u_4(encArg(x_1)) encode_is_NzNat(x_1) -> is_NzNat(encArg(x_1)) encode_True -> True encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_u_11(x_1, x_2, x_3) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_1(x_1, x_2, x_3) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_01(x_1) -> u_01(encArg(x_1)) encode_u_21(x_1, x_2, x_3) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_2(x_1) -> u_2(encArg(x_1)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_u_02(x_1, x_2) -> u_02(encArg(x_1), encArg(x_2)) encode_u_31(x_1, x_2, x_3, x_4) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u_3(x_1, x_2, x_3) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(s(N)) -> N +(N, 0) -> N +(s(N), s(M)) -> s(s(+(N, M))) *(N, 0) -> 0 *(s(N), s(M)) -> s(+(N, +(M, *(N, M)))) gt(0, M) -> False gt(NzN, 0) -> u_4(is_NzNat(NzN)) u_4(True) -> True is_NzNat(0) -> False is_NzNat(s(N)) -> True gt(s(N), s(M)) -> gt(N, M) lt(N, M) -> gt(M, N) d(0, N) -> N d(s(N), s(M)) -> d(N, M) quot(N, NzM) -> u_11(is_NzNat(NzM), N, NzM) u_11(True, N, NzM) -> u_1(gt(N, NzM), N, NzM) u_1(True, N, NzM) -> s(quot(d(N, NzM), NzM)) quot(NzM, NzM) -> u_01(is_NzNat(NzM)) u_01(True) -> s(0) quot(N, NzM) -> u_21(is_NzNat(NzM), NzM, N) u_21(True, NzM, N) -> u_2(gt(NzM, N)) u_2(True) -> 0 gcd(0, N) -> 0 gcd(NzM, NzM) -> u_02(is_NzNat(NzM), NzM) u_02(True, NzM) -> NzM gcd(NzN, NzM) -> u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM) u_31(True, True, NzN, NzM) -> u_3(gt(NzN, NzM), NzN, NzM) u_3(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(False) -> False encArg(True) -> True encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_u_4(x_1)) -> u_4(encArg(x_1)) encArg(cons_is_NzNat(x_1)) -> is_NzNat(encArg(x_1)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_u_11(x_1, x_2, x_3)) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_1(x_1, x_2, x_3)) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_01(x_1)) -> u_01(encArg(x_1)) encArg(cons_u_21(x_1, x_2, x_3)) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_2(x_1)) -> u_2(encArg(x_1)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_u_02(x_1, x_2)) -> u_02(encArg(x_1), encArg(x_2)) encArg(cons_u_31(x_1, x_2, x_3, x_4)) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u_3(x_1, x_2, x_3)) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_False -> False encode_u_4(x_1) -> u_4(encArg(x_1)) encode_is_NzNat(x_1) -> is_NzNat(encArg(x_1)) encode_True -> True encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_u_11(x_1, x_2, x_3) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_1(x_1, x_2, x_3) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_01(x_1) -> u_01(encArg(x_1)) encode_u_21(x_1, x_2, x_3) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_2(x_1) -> u_2(encArg(x_1)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_u_02(x_1, x_2) -> u_02(encArg(x_1), encArg(x_2)) encode_u_31(x_1, x_2, x_3, x_4) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u_3(x_1, x_2, x_3) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(s(N)) -> N +(N, 0) -> N +(s(N), s(M)) -> s(s(+(N, M))) *(N, 0) -> 0 *(s(N), s(M)) -> s(+(N, +(M, *(N, M)))) gt(0, M) -> False gt(NzN, 0) -> u_4(is_NzNat(NzN)) u_4(True) -> True is_NzNat(0) -> False is_NzNat(s(N)) -> True gt(s(N), s(M)) -> gt(N, M) lt(N, M) -> gt(M, N) d(0, N) -> N d(s(N), s(M)) -> d(N, M) quot(N, NzM) -> u_11(is_NzNat(NzM), N, NzM) u_11(True, N, NzM) -> u_1(gt(N, NzM), N, NzM) u_1(True, N, NzM) -> s(quot(d(N, NzM), NzM)) quot(NzM, NzM) -> u_01(is_NzNat(NzM)) u_01(True) -> s(0) quot(N, NzM) -> u_21(is_NzNat(NzM), NzM, N) u_21(True, NzM, N) -> u_2(gt(NzM, N)) u_2(True) -> 0 gcd(0, N) -> 0 gcd(NzM, NzM) -> u_02(is_NzNat(NzM), NzM) u_02(True, NzM) -> NzM gcd(NzN, NzM) -> u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM) u_31(True, True, NzN, NzM) -> u_3(gt(NzN, NzM), NzN, NzM) u_3(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(False) -> False encArg(True) -> True encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_u_4(x_1)) -> u_4(encArg(x_1)) encArg(cons_is_NzNat(x_1)) -> is_NzNat(encArg(x_1)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_u_11(x_1, x_2, x_3)) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_1(x_1, x_2, x_3)) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_01(x_1)) -> u_01(encArg(x_1)) encArg(cons_u_21(x_1, x_2, x_3)) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_2(x_1)) -> u_2(encArg(x_1)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_u_02(x_1, x_2)) -> u_02(encArg(x_1), encArg(x_2)) encArg(cons_u_31(x_1, x_2, x_3, x_4)) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u_3(x_1, x_2, x_3)) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_False -> False encode_u_4(x_1) -> u_4(encArg(x_1)) encode_is_NzNat(x_1) -> is_NzNat(encArg(x_1)) encode_True -> True encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_u_11(x_1, x_2, x_3) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_1(x_1, x_2, x_3) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_01(x_1) -> u_01(encArg(x_1)) encode_u_21(x_1, x_2, x_3) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_2(x_1) -> u_2(encArg(x_1)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_u_02(x_1, x_2) -> u_02(encArg(x_1), encArg(x_2)) encode_u_31(x_1, x_2, x_3, x_4) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u_3(x_1, x_2, x_3) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence gt(s(N), s(M)) ->^+ gt(N, M) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [N / s(N), M / s(M)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) 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, INF). The TRS R consists of the following rules: p(s(N)) -> N +(N, 0) -> N +(s(N), s(M)) -> s(s(+(N, M))) *(N, 0) -> 0 *(s(N), s(M)) -> s(+(N, +(M, *(N, M)))) gt(0, M) -> False gt(NzN, 0) -> u_4(is_NzNat(NzN)) u_4(True) -> True is_NzNat(0) -> False is_NzNat(s(N)) -> True gt(s(N), s(M)) -> gt(N, M) lt(N, M) -> gt(M, N) d(0, N) -> N d(s(N), s(M)) -> d(N, M) quot(N, NzM) -> u_11(is_NzNat(NzM), N, NzM) u_11(True, N, NzM) -> u_1(gt(N, NzM), N, NzM) u_1(True, N, NzM) -> s(quot(d(N, NzM), NzM)) quot(NzM, NzM) -> u_01(is_NzNat(NzM)) u_01(True) -> s(0) quot(N, NzM) -> u_21(is_NzNat(NzM), NzM, N) u_21(True, NzM, N) -> u_2(gt(NzM, N)) u_2(True) -> 0 gcd(0, N) -> 0 gcd(NzM, NzM) -> u_02(is_NzNat(NzM), NzM) u_02(True, NzM) -> NzM gcd(NzN, NzM) -> u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM) u_31(True, True, NzN, NzM) -> u_3(gt(NzN, NzM), NzN, NzM) u_3(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(False) -> False encArg(True) -> True encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_u_4(x_1)) -> u_4(encArg(x_1)) encArg(cons_is_NzNat(x_1)) -> is_NzNat(encArg(x_1)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_u_11(x_1, x_2, x_3)) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_1(x_1, x_2, x_3)) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_01(x_1)) -> u_01(encArg(x_1)) encArg(cons_u_21(x_1, x_2, x_3)) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_2(x_1)) -> u_2(encArg(x_1)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_u_02(x_1, x_2)) -> u_02(encArg(x_1), encArg(x_2)) encArg(cons_u_31(x_1, x_2, x_3, x_4)) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u_3(x_1, x_2, x_3)) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_False -> False encode_u_4(x_1) -> u_4(encArg(x_1)) encode_is_NzNat(x_1) -> is_NzNat(encArg(x_1)) encode_True -> True encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_u_11(x_1, x_2, x_3) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_1(x_1, x_2, x_3) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_01(x_1) -> u_01(encArg(x_1)) encode_u_21(x_1, x_2, x_3) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_2(x_1) -> u_2(encArg(x_1)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_u_02(x_1, x_2) -> u_02(encArg(x_1), encArg(x_2)) encode_u_31(x_1, x_2, x_3, x_4) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u_3(x_1, x_2, x_3) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(s(N)) -> N +(N, 0) -> N +(s(N), s(M)) -> s(s(+(N, M))) *(N, 0) -> 0 *(s(N), s(M)) -> s(+(N, +(M, *(N, M)))) gt(0, M) -> False gt(NzN, 0) -> u_4(is_NzNat(NzN)) u_4(True) -> True is_NzNat(0) -> False is_NzNat(s(N)) -> True gt(s(N), s(M)) -> gt(N, M) lt(N, M) -> gt(M, N) d(0, N) -> N d(s(N), s(M)) -> d(N, M) quot(N, NzM) -> u_11(is_NzNat(NzM), N, NzM) u_11(True, N, NzM) -> u_1(gt(N, NzM), N, NzM) u_1(True, N, NzM) -> s(quot(d(N, NzM), NzM)) quot(NzM, NzM) -> u_01(is_NzNat(NzM)) u_01(True) -> s(0) quot(N, NzM) -> u_21(is_NzNat(NzM), NzM, N) u_21(True, NzM, N) -> u_2(gt(NzM, N)) u_2(True) -> 0 gcd(0, N) -> 0 gcd(NzM, NzM) -> u_02(is_NzNat(NzM), NzM) u_02(True, NzM) -> NzM gcd(NzN, NzM) -> u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM) u_31(True, True, NzN, NzM) -> u_3(gt(NzN, NzM), NzN, NzM) u_3(True, NzN, NzM) -> gcd(d(NzN, NzM), NzM) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(False) -> False encArg(True) -> True encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_u_4(x_1)) -> u_4(encArg(x_1)) encArg(cons_is_NzNat(x_1)) -> is_NzNat(encArg(x_1)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encArg(cons_u_11(x_1, x_2, x_3)) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_1(x_1, x_2, x_3)) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_01(x_1)) -> u_01(encArg(x_1)) encArg(cons_u_21(x_1, x_2, x_3)) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u_2(x_1)) -> u_2(encArg(x_1)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_u_02(x_1, x_2)) -> u_02(encArg(x_1), encArg(x_2)) encArg(cons_u_31(x_1, x_2, x_3, x_4)) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u_3(x_1, x_2, x_3)) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_False -> False encode_u_4(x_1) -> u_4(encArg(x_1)) encode_is_NzNat(x_1) -> is_NzNat(encArg(x_1)) encode_True -> True encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) encode_u_11(x_1, x_2, x_3) -> u_11(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_1(x_1, x_2, x_3) -> u_1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_01(x_1) -> u_01(encArg(x_1)) encode_u_21(x_1, x_2, x_3) -> u_21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u_2(x_1) -> u_2(encArg(x_1)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_u_02(x_1, x_2) -> u_02(encArg(x_1), encArg(x_2)) encode_u_31(x_1, x_2, x_3, x_4) -> u_31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u_3(x_1, x_2, x_3) -> u_3(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST