WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 380 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 4 ms] (6) BEST (7) proven lower bound (8) LowerBoundPropagationProof [FINISHED, 0 ms] (9) BOUNDS(n^1, INF) (10) TRS for Loop Detection ---------------------------------------- (0) 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: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(z0)), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(z0)), z1)), F(S(S(z0)), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 H(0, S(z0)) -> c4(H(0, z0)) H(0, 0) -> c5 H(S(z0), z1) -> c6(H(z0, z1)) G(S(z0), S(z1)) -> c7(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c8(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), G(z0, S(S(z1)))) G(S(z0), 0) -> c9 G(0, z0) -> c10 The (relative) TRS S consists of the following rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(z0)), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(z1)))) g(S(z0), 0) -> 0 g(0, z0) -> 0 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, INF). The TRS R consists of the following rules: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(z0)), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(z0)), z1)), F(S(S(z0)), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 H(0, S(z0)) -> c4(H(0, z0)) H(0, 0) -> c5 H(S(z0), z1) -> c6(H(z0, z1)) G(S(z0), S(z1)) -> c7(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c8(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), G(z0, S(S(z1)))) G(S(z0), 0) -> c9 G(0, z0) -> c10 The (relative) TRS S consists of the following rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(z0)), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(z1)))) g(S(z0), 0) -> 0 g(0, z0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (4) 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: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(z0)), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(z0)), z1)), F(S(S(z0)), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 H(0, S(z0)) -> c4(H(0, z0)) H(0, 0) -> c5 H(S(z0), z1) -> c6(H(z0, z1)) G(S(z0), S(z1)) -> c7(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c8(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), G(z0, S(S(z1)))) G(S(z0), 0) -> c9 G(0, z0) -> c10 The (relative) TRS S consists of the following rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(z0)), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(z1)))) g(S(z0), 0) -> 0 g(0, z0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (5) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence G(S(z0), S(z1)) ->^+ c8(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), G(z0, S(S(z1)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / S(z0)]. The result substitution is [z1 / S(z1)]. ---------------------------------------- (6) Complex Obligation (BEST) ---------------------------------------- (7) 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: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(z0)), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(z0)), z1)), F(S(S(z0)), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 H(0, S(z0)) -> c4(H(0, z0)) H(0, 0) -> c5 H(S(z0), z1) -> c6(H(z0, z1)) G(S(z0), S(z1)) -> c7(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c8(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), G(z0, S(S(z1)))) G(S(z0), 0) -> c9 G(0, z0) -> c10 The (relative) TRS S consists of the following rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(z0)), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(z1)))) g(S(z0), 0) -> 0 g(0, z0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (8) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (9) BOUNDS(n^1, INF) ---------------------------------------- (10) 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: F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(z0)), z1)), G(z0, S(z1))) F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(z0)), z1)), F(S(S(z0)), z1)) F(S(z0), 0) -> c2 F(0, z0) -> c3 H(0, S(z0)) -> c4(H(0, z0)) H(0, 0) -> c5 H(S(z0), z1) -> c6(H(z0, z1)) G(S(z0), S(z1)) -> c7(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), F(S(z0), S(z1))) G(S(z0), S(z1)) -> c8(H(f(S(z0), S(z1)), g(z0, S(S(z1)))), G(z0, S(S(z1)))) G(S(z0), 0) -> c9 G(0, z0) -> c10 The (relative) TRS S consists of the following rules: f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(z0)), z1)) f(S(z0), 0) -> 0 f(0, z0) -> 0 h(0, S(z0)) -> h(0, z0) h(0, 0) -> 0 h(S(z0), z1) -> h(z0, z1) g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(z1)))) g(S(z0), 0) -> 0 g(0, z0) -> 0 Rewrite Strategy: INNERMOST