/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 419 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: t(N) -> cs(r(q(N)), nt(ns(N))) q(0) -> 0 q(s(X)) -> s(p(q(X), d(X))) d(0) -> 0 d(s(X)) -> s(s(d(X))) p(0, X) -> X p(X, 0) -> X p(s(X), s(Y)) -> s(s(p(X, Y))) f(0, X) -> nil f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) t(X) -> nt(X) s(X) -> ns(X) f(X1, X2) -> nf(X1, X2) a(nt(X)) -> t(a(X)) a(ns(X)) -> s(a(X)) a(nf(X1, X2)) -> f(a(X1), a(X2)) a(X) -> X 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(cs(x_1, x_2)) -> cs(encArg(x_1), encArg(x_2)) encArg(r(x_1)) -> r(encArg(x_1)) encArg(nt(x_1)) -> nt(encArg(x_1)) encArg(ns(x_1)) -> ns(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(nf(x_1, x_2)) -> nf(encArg(x_1), encArg(x_2)) encArg(cons_t(x_1)) -> t(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) encode_cs(x_1, x_2) -> cs(encArg(x_1), encArg(x_2)) encode_r(x_1) -> r(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_nt(x_1) -> nt(encArg(x_1)) encode_ns(x_1) -> ns(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2)) encode_d(x_1) -> d(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_nf(x_1, x_2) -> nf(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) ---------------------------------------- (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: t(N) -> cs(r(q(N)), nt(ns(N))) q(0) -> 0 q(s(X)) -> s(p(q(X), d(X))) d(0) -> 0 d(s(X)) -> s(s(d(X))) p(0, X) -> X p(X, 0) -> X p(s(X), s(Y)) -> s(s(p(X, Y))) f(0, X) -> nil f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) t(X) -> nt(X) s(X) -> ns(X) f(X1, X2) -> nf(X1, X2) a(nt(X)) -> t(a(X)) a(ns(X)) -> s(a(X)) a(nf(X1, X2)) -> f(a(X1), a(X2)) a(X) -> X The (relative) TRS S consists of the following rules: encArg(cs(x_1, x_2)) -> cs(encArg(x_1), encArg(x_2)) encArg(r(x_1)) -> r(encArg(x_1)) encArg(nt(x_1)) -> nt(encArg(x_1)) encArg(ns(x_1)) -> ns(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(nf(x_1, x_2)) -> nf(encArg(x_1), encArg(x_2)) encArg(cons_t(x_1)) -> t(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) encode_cs(x_1, x_2) -> cs(encArg(x_1), encArg(x_2)) encode_r(x_1) -> r(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_nt(x_1) -> nt(encArg(x_1)) encode_ns(x_1) -> ns(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2)) encode_d(x_1) -> d(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_nf(x_1, x_2) -> nf(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) 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: t(N) -> cs(r(q(N)), nt(ns(N))) q(0) -> 0 q(s(X)) -> s(p(q(X), d(X))) d(0) -> 0 d(s(X)) -> s(s(d(X))) p(0, X) -> X p(X, 0) -> X p(s(X), s(Y)) -> s(s(p(X, Y))) f(0, X) -> nil f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) t(X) -> nt(X) s(X) -> ns(X) f(X1, X2) -> nf(X1, X2) a(nt(X)) -> t(a(X)) a(ns(X)) -> s(a(X)) a(nf(X1, X2)) -> f(a(X1), a(X2)) a(X) -> X The (relative) TRS S consists of the following rules: encArg(cs(x_1, x_2)) -> cs(encArg(x_1), encArg(x_2)) encArg(r(x_1)) -> r(encArg(x_1)) encArg(nt(x_1)) -> nt(encArg(x_1)) encArg(ns(x_1)) -> ns(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(nf(x_1, x_2)) -> nf(encArg(x_1), encArg(x_2)) encArg(cons_t(x_1)) -> t(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) encode_cs(x_1, x_2) -> cs(encArg(x_1), encArg(x_2)) encode_r(x_1) -> r(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_nt(x_1) -> nt(encArg(x_1)) encode_ns(x_1) -> ns(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2)) encode_d(x_1) -> d(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_nf(x_1, x_2) -> nf(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) 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: t(N) -> cs(r(q(N)), nt(ns(N))) q(0) -> 0 q(s(X)) -> s(p(q(X), d(X))) d(0) -> 0 d(s(X)) -> s(s(d(X))) p(0, X) -> X p(X, 0) -> X p(s(X), s(Y)) -> s(s(p(X, Y))) f(0, X) -> nil f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) t(X) -> nt(X) s(X) -> ns(X) f(X1, X2) -> nf(X1, X2) a(nt(X)) -> t(a(X)) a(ns(X)) -> s(a(X)) a(nf(X1, X2)) -> f(a(X1), a(X2)) a(X) -> X The (relative) TRS S consists of the following rules: encArg(cs(x_1, x_2)) -> cs(encArg(x_1), encArg(x_2)) encArg(r(x_1)) -> r(encArg(x_1)) encArg(nt(x_1)) -> nt(encArg(x_1)) encArg(ns(x_1)) -> ns(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(nf(x_1, x_2)) -> nf(encArg(x_1), encArg(x_2)) encArg(cons_t(x_1)) -> t(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) encode_cs(x_1, x_2) -> cs(encArg(x_1), encArg(x_2)) encode_r(x_1) -> r(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_nt(x_1) -> nt(encArg(x_1)) encode_ns(x_1) -> ns(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2)) encode_d(x_1) -> d(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_nf(x_1, x_2) -> nf(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) 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 a(nt(X)) ->^+ t(a(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / nt(X)]. 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: t(N) -> cs(r(q(N)), nt(ns(N))) q(0) -> 0 q(s(X)) -> s(p(q(X), d(X))) d(0) -> 0 d(s(X)) -> s(s(d(X))) p(0, X) -> X p(X, 0) -> X p(s(X), s(Y)) -> s(s(p(X, Y))) f(0, X) -> nil f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) t(X) -> nt(X) s(X) -> ns(X) f(X1, X2) -> nf(X1, X2) a(nt(X)) -> t(a(X)) a(ns(X)) -> s(a(X)) a(nf(X1, X2)) -> f(a(X1), a(X2)) a(X) -> X The (relative) TRS S consists of the following rules: encArg(cs(x_1, x_2)) -> cs(encArg(x_1), encArg(x_2)) encArg(r(x_1)) -> r(encArg(x_1)) encArg(nt(x_1)) -> nt(encArg(x_1)) encArg(ns(x_1)) -> ns(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(nf(x_1, x_2)) -> nf(encArg(x_1), encArg(x_2)) encArg(cons_t(x_1)) -> t(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) encode_cs(x_1, x_2) -> cs(encArg(x_1), encArg(x_2)) encode_r(x_1) -> r(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_nt(x_1) -> nt(encArg(x_1)) encode_ns(x_1) -> ns(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2)) encode_d(x_1) -> d(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_nf(x_1, x_2) -> nf(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) 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: t(N) -> cs(r(q(N)), nt(ns(N))) q(0) -> 0 q(s(X)) -> s(p(q(X), d(X))) d(0) -> 0 d(s(X)) -> s(s(d(X))) p(0, X) -> X p(X, 0) -> X p(s(X), s(Y)) -> s(s(p(X, Y))) f(0, X) -> nil f(s(X), cs(Y, Z)) -> cs(Y, nf(X, a(Z))) t(X) -> nt(X) s(X) -> ns(X) f(X1, X2) -> nf(X1, X2) a(nt(X)) -> t(a(X)) a(ns(X)) -> s(a(X)) a(nf(X1, X2)) -> f(a(X1), a(X2)) a(X) -> X The (relative) TRS S consists of the following rules: encArg(cs(x_1, x_2)) -> cs(encArg(x_1), encArg(x_2)) encArg(r(x_1)) -> r(encArg(x_1)) encArg(nt(x_1)) -> nt(encArg(x_1)) encArg(ns(x_1)) -> ns(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(nf(x_1, x_2)) -> nf(encArg(x_1), encArg(x_2)) encArg(cons_t(x_1)) -> t(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) encode_cs(x_1, x_2) -> cs(encArg(x_1), encArg(x_2)) encode_r(x_1) -> r(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_nt(x_1) -> nt(encArg(x_1)) encode_ns(x_1) -> ns(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2)) encode_d(x_1) -> d(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_nf(x_1, x_2) -> nf(encArg(x_1), encArg(x_2)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST