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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 1051 ms] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 5 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: A__TERMS(z0) -> c(A__SQR(mark(z0)), MARK(z0)) A__TERMS(z0) -> c1 A__SQR(0) -> c2 A__SQR(s(z0)) -> c3 A__SQR(z0) -> c4 A__DBL(0) -> c5 A__DBL(s(z0)) -> c6 A__DBL(z0) -> c7 A__ADD(0, z0) -> c8(MARK(z0)) A__ADD(s(z0), z1) -> c9 A__ADD(z0, z1) -> c10 A__FIRST(0, z0) -> c11 A__FIRST(s(z0), cons(z1, z2)) -> c12(MARK(z1)) A__FIRST(z0, z1) -> c13 MARK(terms(z0)) -> c14(A__TERMS(mark(z0)), MARK(z0)) MARK(sqr(z0)) -> c15(A__SQR(mark(z0)), MARK(z0)) MARK(add(z0, z1)) -> c16(A__ADD(mark(z0), mark(z1)), MARK(z0)) MARK(add(z0, z1)) -> c17(A__ADD(mark(z0), mark(z1)), MARK(z1)) MARK(dbl(z0)) -> c18(A__DBL(mark(z0)), MARK(z0)) MARK(first(z0, z1)) -> c19(A__FIRST(mark(z0), mark(z1)), MARK(z0)) MARK(first(z0, z1)) -> c20(A__FIRST(mark(z0), mark(z1)), MARK(z1)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(recip(z0)) -> c22(MARK(z0)) MARK(s(z0)) -> c23 MARK(0) -> c24 MARK(nil) -> c25 The (relative) TRS S consists of the following rules: a__terms(z0) -> cons(recip(a__sqr(mark(z0))), terms(s(z0))) a__terms(z0) -> terms(z0) a__sqr(0) -> 0 a__sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) a__sqr(z0) -> sqr(z0) a__dbl(0) -> 0 a__dbl(s(z0)) -> s(s(dbl(z0))) a__dbl(z0) -> dbl(z0) a__add(0, z0) -> mark(z0) a__add(s(z0), z1) -> s(add(z0, z1)) a__add(z0, z1) -> add(z0, z1) a__first(0, z0) -> nil a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2)) a__first(z0, z1) -> first(z0, z1) mark(terms(z0)) -> a__terms(mark(z0)) mark(sqr(z0)) -> a__sqr(mark(z0)) mark(add(z0, z1)) -> a__add(mark(z0), mark(z1)) mark(dbl(z0)) -> a__dbl(mark(z0)) mark(first(z0, z1)) -> a__first(mark(z0), mark(z1)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(recip(z0)) -> recip(mark(z0)) mark(s(z0)) -> s(z0) mark(0) -> 0 mark(nil) -> nil 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__TERMS(z0) -> c(A__SQR(mark(z0)), MARK(z0)) A__TERMS(z0) -> c1 A__SQR(0) -> c2 A__SQR(s(z0)) -> c3 A__SQR(z0) -> c4 A__DBL(0) -> c5 A__DBL(s(z0)) -> c6 A__DBL(z0) -> c7 A__ADD(0, z0) -> c8(MARK(z0)) A__ADD(s(z0), z1) -> c9 A__ADD(z0, z1) -> c10 A__FIRST(0, z0) -> c11 A__FIRST(s(z0), cons(z1, z2)) -> c12(MARK(z1)) A__FIRST(z0, z1) -> c13 MARK(terms(z0)) -> c14(A__TERMS(mark(z0)), MARK(z0)) MARK(sqr(z0)) -> c15(A__SQR(mark(z0)), MARK(z0)) MARK(add(z0, z1)) -> c16(A__ADD(mark(z0), mark(z1)), MARK(z0)) MARK(add(z0, z1)) -> c17(A__ADD(mark(z0), mark(z1)), MARK(z1)) MARK(dbl(z0)) -> c18(A__DBL(mark(z0)), MARK(z0)) MARK(first(z0, z1)) -> c19(A__FIRST(mark(z0), mark(z1)), MARK(z0)) MARK(first(z0, z1)) -> c20(A__FIRST(mark(z0), mark(z1)), MARK(z1)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(recip(z0)) -> c22(MARK(z0)) MARK(s(z0)) -> c23 MARK(0) -> c24 MARK(nil) -> c25 The (relative) TRS S consists of the following rules: a__terms(z0) -> cons(recip(a__sqr(mark(z0))), terms(s(z0))) a__terms(z0) -> terms(z0) a__sqr(0) -> 0 a__sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) a__sqr(z0) -> sqr(z0) a__dbl(0) -> 0 a__dbl(s(z0)) -> s(s(dbl(z0))) a__dbl(z0) -> dbl(z0) a__add(0, z0) -> mark(z0) a__add(s(z0), z1) -> s(add(z0, z1)) a__add(z0, z1) -> add(z0, z1) a__first(0, z0) -> nil a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2)) a__first(z0, z1) -> first(z0, z1) mark(terms(z0)) -> a__terms(mark(z0)) mark(sqr(z0)) -> a__sqr(mark(z0)) mark(add(z0, z1)) -> a__add(mark(z0), mark(z1)) mark(dbl(z0)) -> a__dbl(mark(z0)) mark(first(z0, z1)) -> a__first(mark(z0), mark(z1)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(recip(z0)) -> recip(mark(z0)) mark(s(z0)) -> s(z0) mark(0) -> 0 mark(nil) -> nil 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__TERMS(z0) -> c(A__SQR(mark(z0)), MARK(z0)) A__TERMS(z0) -> c1 A__SQR(0) -> c2 A__SQR(s(z0)) -> c3 A__SQR(z0) -> c4 A__DBL(0) -> c5 A__DBL(s(z0)) -> c6 A__DBL(z0) -> c7 A__ADD(0, z0) -> c8(MARK(z0)) A__ADD(s(z0), z1) -> c9 A__ADD(z0, z1) -> c10 A__FIRST(0, z0) -> c11 A__FIRST(s(z0), cons(z1, z2)) -> c12(MARK(z1)) A__FIRST(z0, z1) -> c13 MARK(terms(z0)) -> c14(A__TERMS(mark(z0)), MARK(z0)) MARK(sqr(z0)) -> c15(A__SQR(mark(z0)), MARK(z0)) MARK(add(z0, z1)) -> c16(A__ADD(mark(z0), mark(z1)), MARK(z0)) MARK(add(z0, z1)) -> c17(A__ADD(mark(z0), mark(z1)), MARK(z1)) MARK(dbl(z0)) -> c18(A__DBL(mark(z0)), MARK(z0)) MARK(first(z0, z1)) -> c19(A__FIRST(mark(z0), mark(z1)), MARK(z0)) MARK(first(z0, z1)) -> c20(A__FIRST(mark(z0), mark(z1)), MARK(z1)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(recip(z0)) -> c22(MARK(z0)) MARK(s(z0)) -> c23 MARK(0) -> c24 MARK(nil) -> c25 The (relative) TRS S consists of the following rules: a__terms(z0) -> cons(recip(a__sqr(mark(z0))), terms(s(z0))) a__terms(z0) -> terms(z0) a__sqr(0) -> 0 a__sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) a__sqr(z0) -> sqr(z0) a__dbl(0) -> 0 a__dbl(s(z0)) -> s(s(dbl(z0))) a__dbl(z0) -> dbl(z0) a__add(0, z0) -> mark(z0) a__add(s(z0), z1) -> s(add(z0, z1)) a__add(z0, z1) -> add(z0, z1) a__first(0, z0) -> nil a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2)) a__first(z0, z1) -> first(z0, z1) mark(terms(z0)) -> a__terms(mark(z0)) mark(sqr(z0)) -> a__sqr(mark(z0)) mark(add(z0, z1)) -> a__add(mark(z0), mark(z1)) mark(dbl(z0)) -> a__dbl(mark(z0)) mark(first(z0, z1)) -> a__first(mark(z0), mark(z1)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(recip(z0)) -> recip(mark(z0)) mark(s(z0)) -> s(z0) mark(0) -> 0 mark(nil) -> nil 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(add(z0, z1)) ->^+ c16(A__ADD(mark(z0), mark(z1)), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / add(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: A__TERMS(z0) -> c(A__SQR(mark(z0)), MARK(z0)) A__TERMS(z0) -> c1 A__SQR(0) -> c2 A__SQR(s(z0)) -> c3 A__SQR(z0) -> c4 A__DBL(0) -> c5 A__DBL(s(z0)) -> c6 A__DBL(z0) -> c7 A__ADD(0, z0) -> c8(MARK(z0)) A__ADD(s(z0), z1) -> c9 A__ADD(z0, z1) -> c10 A__FIRST(0, z0) -> c11 A__FIRST(s(z0), cons(z1, z2)) -> c12(MARK(z1)) A__FIRST(z0, z1) -> c13 MARK(terms(z0)) -> c14(A__TERMS(mark(z0)), MARK(z0)) MARK(sqr(z0)) -> c15(A__SQR(mark(z0)), MARK(z0)) MARK(add(z0, z1)) -> c16(A__ADD(mark(z0), mark(z1)), MARK(z0)) MARK(add(z0, z1)) -> c17(A__ADD(mark(z0), mark(z1)), MARK(z1)) MARK(dbl(z0)) -> c18(A__DBL(mark(z0)), MARK(z0)) MARK(first(z0, z1)) -> c19(A__FIRST(mark(z0), mark(z1)), MARK(z0)) MARK(first(z0, z1)) -> c20(A__FIRST(mark(z0), mark(z1)), MARK(z1)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(recip(z0)) -> c22(MARK(z0)) MARK(s(z0)) -> c23 MARK(0) -> c24 MARK(nil) -> c25 The (relative) TRS S consists of the following rules: a__terms(z0) -> cons(recip(a__sqr(mark(z0))), terms(s(z0))) a__terms(z0) -> terms(z0) a__sqr(0) -> 0 a__sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) a__sqr(z0) -> sqr(z0) a__dbl(0) -> 0 a__dbl(s(z0)) -> s(s(dbl(z0))) a__dbl(z0) -> dbl(z0) a__add(0, z0) -> mark(z0) a__add(s(z0), z1) -> s(add(z0, z1)) a__add(z0, z1) -> add(z0, z1) a__first(0, z0) -> nil a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2)) a__first(z0, z1) -> first(z0, z1) mark(terms(z0)) -> a__terms(mark(z0)) mark(sqr(z0)) -> a__sqr(mark(z0)) mark(add(z0, z1)) -> a__add(mark(z0), mark(z1)) mark(dbl(z0)) -> a__dbl(mark(z0)) mark(first(z0, z1)) -> a__first(mark(z0), mark(z1)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(recip(z0)) -> recip(mark(z0)) mark(s(z0)) -> s(z0) mark(0) -> 0 mark(nil) -> nil 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__TERMS(z0) -> c(A__SQR(mark(z0)), MARK(z0)) A__TERMS(z0) -> c1 A__SQR(0) -> c2 A__SQR(s(z0)) -> c3 A__SQR(z0) -> c4 A__DBL(0) -> c5 A__DBL(s(z0)) -> c6 A__DBL(z0) -> c7 A__ADD(0, z0) -> c8(MARK(z0)) A__ADD(s(z0), z1) -> c9 A__ADD(z0, z1) -> c10 A__FIRST(0, z0) -> c11 A__FIRST(s(z0), cons(z1, z2)) -> c12(MARK(z1)) A__FIRST(z0, z1) -> c13 MARK(terms(z0)) -> c14(A__TERMS(mark(z0)), MARK(z0)) MARK(sqr(z0)) -> c15(A__SQR(mark(z0)), MARK(z0)) MARK(add(z0, z1)) -> c16(A__ADD(mark(z0), mark(z1)), MARK(z0)) MARK(add(z0, z1)) -> c17(A__ADD(mark(z0), mark(z1)), MARK(z1)) MARK(dbl(z0)) -> c18(A__DBL(mark(z0)), MARK(z0)) MARK(first(z0, z1)) -> c19(A__FIRST(mark(z0), mark(z1)), MARK(z0)) MARK(first(z0, z1)) -> c20(A__FIRST(mark(z0), mark(z1)), MARK(z1)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(recip(z0)) -> c22(MARK(z0)) MARK(s(z0)) -> c23 MARK(0) -> c24 MARK(nil) -> c25 The (relative) TRS S consists of the following rules: a__terms(z0) -> cons(recip(a__sqr(mark(z0))), terms(s(z0))) a__terms(z0) -> terms(z0) a__sqr(0) -> 0 a__sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) a__sqr(z0) -> sqr(z0) a__dbl(0) -> 0 a__dbl(s(z0)) -> s(s(dbl(z0))) a__dbl(z0) -> dbl(z0) a__add(0, z0) -> mark(z0) a__add(s(z0), z1) -> s(add(z0, z1)) a__add(z0, z1) -> add(z0, z1) a__first(0, z0) -> nil a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2)) a__first(z0, z1) -> first(z0, z1) mark(terms(z0)) -> a__terms(mark(z0)) mark(sqr(z0)) -> a__sqr(mark(z0)) mark(add(z0, z1)) -> a__add(mark(z0), mark(z1)) mark(dbl(z0)) -> a__dbl(mark(z0)) mark(first(z0, z1)) -> a__first(mark(z0), mark(z1)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(recip(z0)) -> recip(mark(z0)) mark(s(z0)) -> s(z0) mark(0) -> 0 mark(nil) -> nil Rewrite Strategy: INNERMOST