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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 421 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 1 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: A__F(z0) -> c1(A__IF(mark(z0), c, f(true)), MARK(z0)) A__F(z0) -> c2 A__IF(true, z0, z1) -> c3(MARK(z0)) A__IF(false, z0, z1) -> c4(MARK(z1)) A__IF(z0, z1, z2) -> c5 MARK(f(z0)) -> c6(A__F(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c7(A__IF(mark(z0), mark(z1), z2), MARK(z0)) MARK(if(z0, z1, z2)) -> c8(A__IF(mark(z0), mark(z1), z2), MARK(z1)) MARK(c) -> c9 MARK(true) -> c10 MARK(false) -> c11 The (relative) TRS S consists of the following rules: a__f(z0) -> a__if(mark(z0), c, f(true)) a__f(z0) -> f(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(f(z0)) -> a__f(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), mark(z1), z2) mark(c) -> c mark(true) -> true mark(false) -> false 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: A__F(z0) -> c1(A__IF(mark(z0), c, f(true)), MARK(z0)) A__F(z0) -> c2 A__IF(true, z0, z1) -> c3(MARK(z0)) A__IF(false, z0, z1) -> c4(MARK(z1)) A__IF(z0, z1, z2) -> c5 MARK(f(z0)) -> c6(A__F(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c7(A__IF(mark(z0), mark(z1), z2), MARK(z0)) MARK(if(z0, z1, z2)) -> c8(A__IF(mark(z0), mark(z1), z2), MARK(z1)) MARK(c) -> c9 MARK(true) -> c10 MARK(false) -> c11 The (relative) TRS S consists of the following rules: a__f(z0) -> a__if(mark(z0), c, f(true)) a__f(z0) -> f(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(f(z0)) -> a__f(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), mark(z1), z2) mark(c) -> c mark(true) -> true mark(false) -> false 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: A__F(z0) -> c1(A__IF(mark(z0), c, f(true)), MARK(z0)) A__F(z0) -> c2 A__IF(true, z0, z1) -> c3(MARK(z0)) A__IF(false, z0, z1) -> c4(MARK(z1)) A__IF(z0, z1, z2) -> c5 MARK(f(z0)) -> c6(A__F(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c7(A__IF(mark(z0), mark(z1), z2), MARK(z0)) MARK(if(z0, z1, z2)) -> c8(A__IF(mark(z0), mark(z1), z2), MARK(z1)) MARK(c) -> c9 MARK(true) -> c10 MARK(false) -> c11 The (relative) TRS S consists of the following rules: a__f(z0) -> a__if(mark(z0), c, f(true)) a__f(z0) -> f(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(f(z0)) -> a__f(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), mark(z1), z2) mark(c) -> c mark(true) -> true mark(false) -> false 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 MARK(if(z0, z1, z2)) ->^+ c8(A__IF(mark(z0), mark(z1), z2), MARK(z1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z1 / if(z0, z1, z2)]. 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: A__F(z0) -> c1(A__IF(mark(z0), c, f(true)), MARK(z0)) A__F(z0) -> c2 A__IF(true, z0, z1) -> c3(MARK(z0)) A__IF(false, z0, z1) -> c4(MARK(z1)) A__IF(z0, z1, z2) -> c5 MARK(f(z0)) -> c6(A__F(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c7(A__IF(mark(z0), mark(z1), z2), MARK(z0)) MARK(if(z0, z1, z2)) -> c8(A__IF(mark(z0), mark(z1), z2), MARK(z1)) MARK(c) -> c9 MARK(true) -> c10 MARK(false) -> c11 The (relative) TRS S consists of the following rules: a__f(z0) -> a__if(mark(z0), c, f(true)) a__f(z0) -> f(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(f(z0)) -> a__f(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), mark(z1), z2) mark(c) -> c mark(true) -> true mark(false) -> false 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: A__F(z0) -> c1(A__IF(mark(z0), c, f(true)), MARK(z0)) A__F(z0) -> c2 A__IF(true, z0, z1) -> c3(MARK(z0)) A__IF(false, z0, z1) -> c4(MARK(z1)) A__IF(z0, z1, z2) -> c5 MARK(f(z0)) -> c6(A__F(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c7(A__IF(mark(z0), mark(z1), z2), MARK(z0)) MARK(if(z0, z1, z2)) -> c8(A__IF(mark(z0), mark(z1), z2), MARK(z1)) MARK(c) -> c9 MARK(true) -> c10 MARK(false) -> c11 The (relative) TRS S consists of the following rules: a__f(z0) -> a__if(mark(z0), c, f(true)) a__f(z0) -> f(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(f(z0)) -> a__f(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), mark(z1), z2) mark(c) -> c mark(true) -> true mark(false) -> false Rewrite Strategy: INNERMOST