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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 960 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: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) The (relative) TRS S consists of the following rules: <'(S(z0), S(z1)) -> c(<'(z0, z1)) <'(0, S(z0)) -> c1 <'(z0, 0) -> c2 >'(S(z0), S(z1)) -> c3(>'(z0, z1)) >'(0, z0) -> c4 >'(S(z0), 0) -> c5 PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(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: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) The (relative) TRS S consists of the following rules: <'(S(z0), S(z1)) -> c(<'(z0, z1)) <'(0, S(z0)) -> c1 <'(z0, 0) -> c2 >'(S(z0), S(z1)) -> c3(>'(z0, z1)) >'(0, z0) -> c4 >'(S(z0), 0) -> c5 PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(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: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) The (relative) TRS S consists of the following rules: <'(S(z0), S(z1)) -> c(<'(z0, z1)) <'(0, S(z0)) -> c1 <'(z0, 0) -> c2 >'(S(z0), S(z1)) -> c3(>'(z0, z1)) >'(0, z0) -> c4 >'(S(z0), 0) -> c5 PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(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 APP(Cons(z0, z1), z2) ->^+ c17(APP(z1, z2)) 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: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) The (relative) TRS S consists of the following rules: <'(S(z0), S(z1)) -> c(<'(z0, z1)) <'(0, S(z0)) -> c1 <'(z0, 0) -> c2 >'(S(z0), S(z1)) -> c3(>'(z0, z1)) >'(0, z0) -> c4 >'(S(z0), 0) -> c5 PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(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: QUICKSORT(Cons(z0, Cons(z1, z2))) -> c10(PART(z0, Cons(z1, z2))) QUICKSORT(Cons(z0, Nil)) -> c11 QUICKSORT(Nil) -> c12 PARTLT(z0, Cons(z1, z2)) -> c13(PARTLT[ITE][TRUE][ITE](<(z1, z0), z0, Cons(z1, z2)), <'(z1, z0)) PARTLT(z0, Nil) -> c14 PARTGT(z0, Cons(z1, z2)) -> c15(PARTGT[ITE][TRUE][ITE](>(z1, z0), z0, Cons(z1, z2)), >'(z1, z0)) PARTGT(z0, Nil) -> c16 APP(Cons(z0, z1), z2) -> c17(APP(z1, z2)) APP(Nil, z0) -> c18 NOTEMPTY(Cons(z0, z1)) -> c19 NOTEMPTY(Nil) -> c20 PART(z0, z1) -> c21(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partLt(z0, z1)), PARTLT(z0, z1)) PART(z0, z1) -> c22(APP(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))), QUICKSORT(partGt(z0, z1)), PARTGT(z0, z1)) GOAL(z0) -> c23(QUICKSORT(z0)) The (relative) TRS S consists of the following rules: <'(S(z0), S(z1)) -> c(<'(z0, z1)) <'(0, S(z0)) -> c1 <'(z0, 0) -> c2 >'(S(z0), S(z1)) -> c3(>'(z0, z1)) >'(0, z0) -> c4 >'(S(z0), 0) -> c5 PARTLT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c6(PARTLT(z0, z2)) PARTLT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c7(PARTLT(z0, z2)) PARTGT[ITE][TRUE][ITE](True, z0, Cons(z1, z2)) -> c8(PARTGT(z0, z2)) PARTGT[ITE][TRUE][ITE](False, z0, Cons(z1, z2)) -> c9(PARTGT(z0, z2)) <(S(z0), S(z1)) -> <(z0, z1) <(0, S(z0)) -> True <(z0, 0) -> False >(S(z0), S(z1)) -> >(z0, z1) >(0, z0) -> False >(S(z0), 0) -> True partLt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partLt(z0, z2)) partLt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partLt(z0, z2) partGt[Ite][True][Ite](True, z0, Cons(z1, z2)) -> Cons(z1, partGt(z0, z2)) partGt[Ite][True][Ite](False, z0, Cons(z1, z2)) -> partGt(z0, z2) quicksort(Cons(z0, Cons(z1, z2))) -> part(z0, Cons(z1, z2)) quicksort(Cons(z0, Nil)) -> Cons(z0, Nil) quicksort(Nil) -> Nil partLt(z0, Cons(z1, z2)) -> partLt[Ite][True][Ite](<(z1, z0), z0, Cons(z1, z2)) partLt(z0, Nil) -> Nil partGt(z0, Cons(z1, z2)) -> partGt[Ite][True][Ite](>(z1, z0), z0, Cons(z1, z2)) partGt(z0, Nil) -> Nil app(Cons(z0, z1), z2) -> Cons(z0, app(z1, z2)) app(Nil, z0) -> z0 notEmpty(Cons(z0, z1)) -> True notEmpty(Nil) -> False part(z0, z1) -> app(quicksort(partLt(z0, z1)), Cons(z0, quicksort(partGt(z0, z1)))) goal(z0) -> quicksort(z0) Rewrite Strategy: INNERMOST