WORST_CASE(Omega(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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 301 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 0 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: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1 FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) ADD(z0, z1) -> c5 SEL(0, cons(z0, z1)) -> c6 SEL(s(z0), cons(z1, z2)) -> c7(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c8(FIB1(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__fib1(z0, z1)) -> c9(FIB1(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(n__add(z0, z1)) -> c10(ADD(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__add(z0, z1)) -> c11(ADD(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(z0) -> c12 The (relative) TRS S consists of the following rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, n__add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) add(z0, z1) -> n__add(z0, z1) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(activate(z0), activate(z1)) activate(n__add(z0, z1)) -> add(activate(z0), activate(z1)) activate(z0) -> z0 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: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1 FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) ADD(z0, z1) -> c5 SEL(0, cons(z0, z1)) -> c6 SEL(s(z0), cons(z1, z2)) -> c7(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c8(FIB1(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__fib1(z0, z1)) -> c9(FIB1(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(n__add(z0, z1)) -> c10(ADD(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__add(z0, z1)) -> c11(ADD(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(z0) -> c12 The (relative) TRS S consists of the following rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, n__add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) add(z0, z1) -> n__add(z0, z1) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(activate(z0), activate(z1)) activate(n__add(z0, z1)) -> add(activate(z0), activate(z1)) activate(z0) -> z0 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: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1 FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) ADD(z0, z1) -> c5 SEL(0, cons(z0, z1)) -> c6 SEL(s(z0), cons(z1, z2)) -> c7(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c8(FIB1(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__fib1(z0, z1)) -> c9(FIB1(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(n__add(z0, z1)) -> c10(ADD(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__add(z0, z1)) -> c11(ADD(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(z0) -> c12 The (relative) TRS S consists of the following rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, n__add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) add(z0, z1) -> n__add(z0, z1) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(activate(z0), activate(z1)) activate(n__add(z0, z1)) -> add(activate(z0), activate(z1)) activate(z0) -> z0 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 ACTIVATE(n__fib1(z0, z1)) ->^+ c9(FIB1(activate(z0), activate(z1)), ACTIVATE(z1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z1 / n__fib1(z0, z1)]. The result substitution is [ ]. ---------------------------------------- (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: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1 FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) ADD(z0, z1) -> c5 SEL(0, cons(z0, z1)) -> c6 SEL(s(z0), cons(z1, z2)) -> c7(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c8(FIB1(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__fib1(z0, z1)) -> c9(FIB1(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(n__add(z0, z1)) -> c10(ADD(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__add(z0, z1)) -> c11(ADD(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(z0) -> c12 The (relative) TRS S consists of the following rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, n__add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) add(z0, z1) -> n__add(z0, z1) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(activate(z0), activate(z1)) activate(n__add(z0, z1)) -> add(activate(z0), activate(z1)) activate(z0) -> z0 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: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1 FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) ADD(z0, z1) -> c5 SEL(0, cons(z0, z1)) -> c6 SEL(s(z0), cons(z1, z2)) -> c7(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c8(FIB1(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__fib1(z0, z1)) -> c9(FIB1(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(n__add(z0, z1)) -> c10(ADD(activate(z0), activate(z1)), ACTIVATE(z0)) ACTIVATE(n__add(z0, z1)) -> c11(ADD(activate(z0), activate(z1)), ACTIVATE(z1)) ACTIVATE(z0) -> c12 The (relative) TRS S consists of the following rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, n__add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) add(z0, z1) -> n__add(z0, z1) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(activate(z0), activate(z1)) activate(n__add(z0, z1)) -> add(activate(z0), activate(z1)) activate(z0) -> z0 Rewrite Strategy: INNERMOST