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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 1859 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: APP(z0, z1) -> c(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z0)) APP(z0, z1) -> c1(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z1)) PLUS(z0, 0) -> c2 PLUS(z0, s(z1)) -> c3(PLUS(z0, z1)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(LENGTH(z1)) HELPA(z0, z1, z2, z3) -> c6(IF(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1)) GE(z0, 0) -> c7 GE(0, s(z0)) -> c8 GE(s(z0), s(z1)) -> c9(GE(z0, z1)) IF(true, z0, z1, z2, z3) -> c10 IF(false, z0, z1, z2, z3) -> c11(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), GREATER(z2, z3)) IF(false, z0, z1, z2, z3) -> c12(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), SMALLER(z2, z3)) GREATER(z0, z1) -> c13(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z0)) GREATER(z0, z1) -> c14(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z1)) SMALLER(z0, z1) -> c15(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z0)) SMALLER(z0, z1) -> c16(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z1)) HELPC(true, z0, z1) -> c17 HELPC(false, z0, z1) -> c18 HELPB(z0, z1, cons(z2, z3), z4) -> c19(HELPA(s(z0), z1, z3, z4)) The (relative) TRS S consists of the following rules: app(z0, z1) -> helpa(0, plus(length(z0), length(z1)), z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) helpa(z0, z1, z2, z3) -> if(ge(z0, z1), z0, z1, z2, z3) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if(true, z0, z1, z2, z3) -> nil if(false, z0, z1, z2, z3) -> helpb(z0, z1, greater(z2, z3), smaller(z2, z3)) greater(z0, z1) -> helpc(ge(length(z0), length(z1)), z0, z1) smaller(z0, z1) -> helpc(ge(length(z0), length(z1)), z1, z0) helpc(true, z0, z1) -> z0 helpc(false, z0, z1) -> z1 helpb(z0, z1, cons(z2, z3), z4) -> cons(z2, helpa(s(z0), z1, z3, z4)) 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: APP(z0, z1) -> c(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z0)) APP(z0, z1) -> c1(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z1)) PLUS(z0, 0) -> c2 PLUS(z0, s(z1)) -> c3(PLUS(z0, z1)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(LENGTH(z1)) HELPA(z0, z1, z2, z3) -> c6(IF(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1)) GE(z0, 0) -> c7 GE(0, s(z0)) -> c8 GE(s(z0), s(z1)) -> c9(GE(z0, z1)) IF(true, z0, z1, z2, z3) -> c10 IF(false, z0, z1, z2, z3) -> c11(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), GREATER(z2, z3)) IF(false, z0, z1, z2, z3) -> c12(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), SMALLER(z2, z3)) GREATER(z0, z1) -> c13(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z0)) GREATER(z0, z1) -> c14(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z1)) SMALLER(z0, z1) -> c15(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z0)) SMALLER(z0, z1) -> c16(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z1)) HELPC(true, z0, z1) -> c17 HELPC(false, z0, z1) -> c18 HELPB(z0, z1, cons(z2, z3), z4) -> c19(HELPA(s(z0), z1, z3, z4)) The (relative) TRS S consists of the following rules: app(z0, z1) -> helpa(0, plus(length(z0), length(z1)), z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) helpa(z0, z1, z2, z3) -> if(ge(z0, z1), z0, z1, z2, z3) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if(true, z0, z1, z2, z3) -> nil if(false, z0, z1, z2, z3) -> helpb(z0, z1, greater(z2, z3), smaller(z2, z3)) greater(z0, z1) -> helpc(ge(length(z0), length(z1)), z0, z1) smaller(z0, z1) -> helpc(ge(length(z0), length(z1)), z1, z0) helpc(true, z0, z1) -> z0 helpc(false, z0, z1) -> z1 helpb(z0, z1, cons(z2, z3), z4) -> cons(z2, helpa(s(z0), z1, z3, z4)) 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: APP(z0, z1) -> c(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z0)) APP(z0, z1) -> c1(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z1)) PLUS(z0, 0) -> c2 PLUS(z0, s(z1)) -> c3(PLUS(z0, z1)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(LENGTH(z1)) HELPA(z0, z1, z2, z3) -> c6(IF(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1)) GE(z0, 0) -> c7 GE(0, s(z0)) -> c8 GE(s(z0), s(z1)) -> c9(GE(z0, z1)) IF(true, z0, z1, z2, z3) -> c10 IF(false, z0, z1, z2, z3) -> c11(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), GREATER(z2, z3)) IF(false, z0, z1, z2, z3) -> c12(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), SMALLER(z2, z3)) GREATER(z0, z1) -> c13(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z0)) GREATER(z0, z1) -> c14(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z1)) SMALLER(z0, z1) -> c15(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z0)) SMALLER(z0, z1) -> c16(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z1)) HELPC(true, z0, z1) -> c17 HELPC(false, z0, z1) -> c18 HELPB(z0, z1, cons(z2, z3), z4) -> c19(HELPA(s(z0), z1, z3, z4)) The (relative) TRS S consists of the following rules: app(z0, z1) -> helpa(0, plus(length(z0), length(z1)), z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) helpa(z0, z1, z2, z3) -> if(ge(z0, z1), z0, z1, z2, z3) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if(true, z0, z1, z2, z3) -> nil if(false, z0, z1, z2, z3) -> helpb(z0, z1, greater(z2, z3), smaller(z2, z3)) greater(z0, z1) -> helpc(ge(length(z0), length(z1)), z0, z1) smaller(z0, z1) -> helpc(ge(length(z0), length(z1)), z1, z0) helpc(true, z0, z1) -> z0 helpc(false, z0, z1) -> z1 helpb(z0, z1, cons(z2, z3), z4) -> cons(z2, helpa(s(z0), z1, z3, z4)) 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 LENGTH(cons(z0, z1)) ->^+ c5(LENGTH(z1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [z1 / cons(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: APP(z0, z1) -> c(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z0)) APP(z0, z1) -> c1(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z1)) PLUS(z0, 0) -> c2 PLUS(z0, s(z1)) -> c3(PLUS(z0, z1)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(LENGTH(z1)) HELPA(z0, z1, z2, z3) -> c6(IF(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1)) GE(z0, 0) -> c7 GE(0, s(z0)) -> c8 GE(s(z0), s(z1)) -> c9(GE(z0, z1)) IF(true, z0, z1, z2, z3) -> c10 IF(false, z0, z1, z2, z3) -> c11(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), GREATER(z2, z3)) IF(false, z0, z1, z2, z3) -> c12(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), SMALLER(z2, z3)) GREATER(z0, z1) -> c13(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z0)) GREATER(z0, z1) -> c14(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z1)) SMALLER(z0, z1) -> c15(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z0)) SMALLER(z0, z1) -> c16(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z1)) HELPC(true, z0, z1) -> c17 HELPC(false, z0, z1) -> c18 HELPB(z0, z1, cons(z2, z3), z4) -> c19(HELPA(s(z0), z1, z3, z4)) The (relative) TRS S consists of the following rules: app(z0, z1) -> helpa(0, plus(length(z0), length(z1)), z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) helpa(z0, z1, z2, z3) -> if(ge(z0, z1), z0, z1, z2, z3) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if(true, z0, z1, z2, z3) -> nil if(false, z0, z1, z2, z3) -> helpb(z0, z1, greater(z2, z3), smaller(z2, z3)) greater(z0, z1) -> helpc(ge(length(z0), length(z1)), z0, z1) smaller(z0, z1) -> helpc(ge(length(z0), length(z1)), z1, z0) helpc(true, z0, z1) -> z0 helpc(false, z0, z1) -> z1 helpb(z0, z1, cons(z2, z3), z4) -> cons(z2, helpa(s(z0), z1, z3, z4)) 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: APP(z0, z1) -> c(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z0)) APP(z0, z1) -> c1(HELPA(0, plus(length(z0), length(z1)), z0, z1), PLUS(length(z0), length(z1)), LENGTH(z1)) PLUS(z0, 0) -> c2 PLUS(z0, s(z1)) -> c3(PLUS(z0, z1)) LENGTH(nil) -> c4 LENGTH(cons(z0, z1)) -> c5(LENGTH(z1)) HELPA(z0, z1, z2, z3) -> c6(IF(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1)) GE(z0, 0) -> c7 GE(0, s(z0)) -> c8 GE(s(z0), s(z1)) -> c9(GE(z0, z1)) IF(true, z0, z1, z2, z3) -> c10 IF(false, z0, z1, z2, z3) -> c11(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), GREATER(z2, z3)) IF(false, z0, z1, z2, z3) -> c12(HELPB(z0, z1, greater(z2, z3), smaller(z2, z3)), SMALLER(z2, z3)) GREATER(z0, z1) -> c13(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z0)) GREATER(z0, z1) -> c14(HELPC(ge(length(z0), length(z1)), z0, z1), GE(length(z0), length(z1)), LENGTH(z1)) SMALLER(z0, z1) -> c15(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z0)) SMALLER(z0, z1) -> c16(HELPC(ge(length(z0), length(z1)), z1, z0), GE(length(z0), length(z1)), LENGTH(z1)) HELPC(true, z0, z1) -> c17 HELPC(false, z0, z1) -> c18 HELPB(z0, z1, cons(z2, z3), z4) -> c19(HELPA(s(z0), z1, z3, z4)) The (relative) TRS S consists of the following rules: app(z0, z1) -> helpa(0, plus(length(z0), length(z1)), z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) helpa(z0, z1, z2, z3) -> if(ge(z0, z1), z0, z1, z2, z3) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) if(true, z0, z1, z2, z3) -> nil if(false, z0, z1, z2, z3) -> helpb(z0, z1, greater(z2, z3), smaller(z2, z3)) greater(z0, z1) -> helpc(ge(length(z0), length(z1)), z0, z1) smaller(z0, z1) -> helpc(ge(length(z0), length(z1)), z1, z0) helpc(true, z0, z1) -> z0 helpc(false, z0, z1) -> z1 helpb(z0, z1, cons(z2, z3), z4) -> cons(z2, helpa(s(z0), z1, z3, z4)) Rewrite Strategy: INNERMOST