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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).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), 114.8 s] (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: A__FROM(z0) -> c(MARK(z0)) A__FROM(z0) -> c1 A__2NDSPOS(0, z0) -> c2 A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c3(MARK(z2)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c4(A__2NDSNEG(mark(z0), mark(z3)), MARK(z0)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c5(A__2NDSNEG(mark(z0), mark(z3)), MARK(z3)) A__2NDSPOS(z0, z1) -> c6 A__2NDSNEG(0, z0) -> c7 A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c8(MARK(z2)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c9(A__2NDSPOS(mark(z0), mark(z3)), MARK(z0)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c10(A__2NDSPOS(mark(z0), mark(z3)), MARK(z3)) A__2NDSNEG(z0, z1) -> c11 A__PI(z0) -> c12(A__2NDSPOS(mark(z0), a__from(0)), MARK(z0)) A__PI(z0) -> c13(A__2NDSPOS(mark(z0), a__from(0)), A__FROM(0)) A__PI(z0) -> c14 A__PLUS(0, z0) -> c15(MARK(z0)) A__PLUS(s(z0), z1) -> c16(A__PLUS(mark(z0), mark(z1)), MARK(z0)) A__PLUS(s(z0), z1) -> c17(A__PLUS(mark(z0), mark(z1)), MARK(z1)) A__PLUS(z0, z1) -> c18 A__TIMES(0, z0) -> c19 A__TIMES(s(z0), z1) -> c20(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), MARK(z1)) A__TIMES(s(z0), z1) -> c21(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z0)) A__TIMES(s(z0), z1) -> c22(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z1)) A__TIMES(z0, z1) -> c23 A__SQUARE(z0) -> c24(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c25(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c26 MARK(from(z0)) -> c27(A__FROM(mark(z0)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c28(A__2NDSPOS(mark(z0), mark(z1)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c29(A__2NDSPOS(mark(z0), mark(z1)), MARK(z1)) MARK(2ndsneg(z0, z1)) -> c30(A__2NDSNEG(mark(z0), mark(z1)), MARK(z0)) MARK(2ndsneg(z0, z1)) -> c31(A__2NDSNEG(mark(z0), mark(z1)), MARK(z1)) MARK(pi(z0)) -> c32(A__PI(mark(z0)), MARK(z0)) MARK(plus(z0, z1)) -> c33(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c34(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(times(z0, z1)) -> c35(A__TIMES(mark(z0), mark(z1)), MARK(z0)) MARK(times(z0, z1)) -> c36(A__TIMES(mark(z0), mark(z1)), MARK(z1)) MARK(square(z0)) -> c37(A__SQUARE(mark(z0)), MARK(z0)) MARK(0) -> c38 MARK(s(z0)) -> c39(MARK(z0)) MARK(posrecip(z0)) -> c40(MARK(z0)) MARK(negrecip(z0)) -> c41(MARK(z0)) MARK(nil) -> c42 MARK(cons(z0, z1)) -> c43(MARK(z0)) MARK(rnil) -> c44 MARK(rcons(z0, z1)) -> c45(MARK(z0)) MARK(rcons(z0, z1)) -> c46(MARK(z1)) The (relative) TRS S consists of the following rules: a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__2ndspos(0, z0) -> rnil a__2ndspos(s(z0), cons(z1, cons(z2, z3))) -> rcons(posrecip(mark(z2)), a__2ndsneg(mark(z0), mark(z3))) a__2ndspos(z0, z1) -> 2ndspos(z0, z1) a__2ndsneg(0, z0) -> rnil a__2ndsneg(s(z0), cons(z1, cons(z2, z3))) -> rcons(negrecip(mark(z2)), a__2ndspos(mark(z0), mark(z3))) a__2ndsneg(z0, z1) -> 2ndsneg(z0, z1) a__pi(z0) -> a__2ndspos(mark(z0), a__from(0)) a__pi(z0) -> pi(z0) a__plus(0, z0) -> mark(z0) a__plus(s(z0), z1) -> s(a__plus(mark(z0), mark(z1))) a__plus(z0, z1) -> plus(z0, z1) a__times(0, z0) -> 0 a__times(s(z0), z1) -> a__plus(mark(z1), a__times(mark(z0), mark(z1))) a__times(z0, z1) -> times(z0, z1) a__square(z0) -> a__times(mark(z0), mark(z0)) a__square(z0) -> square(z0) mark(from(z0)) -> a__from(mark(z0)) mark(2ndspos(z0, z1)) -> a__2ndspos(mark(z0), mark(z1)) mark(2ndsneg(z0, z1)) -> a__2ndsneg(mark(z0), mark(z1)) mark(pi(z0)) -> a__pi(mark(z0)) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(times(z0, z1)) -> a__times(mark(z0), mark(z1)) mark(square(z0)) -> a__square(mark(z0)) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(posrecip(z0)) -> posrecip(mark(z0)) mark(negrecip(z0)) -> negrecip(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(rnil) -> rnil mark(rcons(z0, z1)) -> rcons(mark(z0), mark(z1)) 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__FROM(z0) -> c(MARK(z0)) A__FROM(z0) -> c1 A__2NDSPOS(0, z0) -> c2 A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c3(MARK(z2)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c4(A__2NDSNEG(mark(z0), mark(z3)), MARK(z0)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c5(A__2NDSNEG(mark(z0), mark(z3)), MARK(z3)) A__2NDSPOS(z0, z1) -> c6 A__2NDSNEG(0, z0) -> c7 A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c8(MARK(z2)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c9(A__2NDSPOS(mark(z0), mark(z3)), MARK(z0)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c10(A__2NDSPOS(mark(z0), mark(z3)), MARK(z3)) A__2NDSNEG(z0, z1) -> c11 A__PI(z0) -> c12(A__2NDSPOS(mark(z0), a__from(0)), MARK(z0)) A__PI(z0) -> c13(A__2NDSPOS(mark(z0), a__from(0)), A__FROM(0)) A__PI(z0) -> c14 A__PLUS(0, z0) -> c15(MARK(z0)) A__PLUS(s(z0), z1) -> c16(A__PLUS(mark(z0), mark(z1)), MARK(z0)) A__PLUS(s(z0), z1) -> c17(A__PLUS(mark(z0), mark(z1)), MARK(z1)) A__PLUS(z0, z1) -> c18 A__TIMES(0, z0) -> c19 A__TIMES(s(z0), z1) -> c20(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), MARK(z1)) A__TIMES(s(z0), z1) -> c21(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z0)) A__TIMES(s(z0), z1) -> c22(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z1)) A__TIMES(z0, z1) -> c23 A__SQUARE(z0) -> c24(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c25(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c26 MARK(from(z0)) -> c27(A__FROM(mark(z0)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c28(A__2NDSPOS(mark(z0), mark(z1)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c29(A__2NDSPOS(mark(z0), mark(z1)), MARK(z1)) MARK(2ndsneg(z0, z1)) -> c30(A__2NDSNEG(mark(z0), mark(z1)), MARK(z0)) MARK(2ndsneg(z0, z1)) -> c31(A__2NDSNEG(mark(z0), mark(z1)), MARK(z1)) MARK(pi(z0)) -> c32(A__PI(mark(z0)), MARK(z0)) MARK(plus(z0, z1)) -> c33(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c34(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(times(z0, z1)) -> c35(A__TIMES(mark(z0), mark(z1)), MARK(z0)) MARK(times(z0, z1)) -> c36(A__TIMES(mark(z0), mark(z1)), MARK(z1)) MARK(square(z0)) -> c37(A__SQUARE(mark(z0)), MARK(z0)) MARK(0) -> c38 MARK(s(z0)) -> c39(MARK(z0)) MARK(posrecip(z0)) -> c40(MARK(z0)) MARK(negrecip(z0)) -> c41(MARK(z0)) MARK(nil) -> c42 MARK(cons(z0, z1)) -> c43(MARK(z0)) MARK(rnil) -> c44 MARK(rcons(z0, z1)) -> c45(MARK(z0)) MARK(rcons(z0, z1)) -> c46(MARK(z1)) The (relative) TRS S consists of the following rules: a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__2ndspos(0, z0) -> rnil a__2ndspos(s(z0), cons(z1, cons(z2, z3))) -> rcons(posrecip(mark(z2)), a__2ndsneg(mark(z0), mark(z3))) a__2ndspos(z0, z1) -> 2ndspos(z0, z1) a__2ndsneg(0, z0) -> rnil a__2ndsneg(s(z0), cons(z1, cons(z2, z3))) -> rcons(negrecip(mark(z2)), a__2ndspos(mark(z0), mark(z3))) a__2ndsneg(z0, z1) -> 2ndsneg(z0, z1) a__pi(z0) -> a__2ndspos(mark(z0), a__from(0)) a__pi(z0) -> pi(z0) a__plus(0, z0) -> mark(z0) a__plus(s(z0), z1) -> s(a__plus(mark(z0), mark(z1))) a__plus(z0, z1) -> plus(z0, z1) a__times(0, z0) -> 0 a__times(s(z0), z1) -> a__plus(mark(z1), a__times(mark(z0), mark(z1))) a__times(z0, z1) -> times(z0, z1) a__square(z0) -> a__times(mark(z0), mark(z0)) a__square(z0) -> square(z0) mark(from(z0)) -> a__from(mark(z0)) mark(2ndspos(z0, z1)) -> a__2ndspos(mark(z0), mark(z1)) mark(2ndsneg(z0, z1)) -> a__2ndsneg(mark(z0), mark(z1)) mark(pi(z0)) -> a__pi(mark(z0)) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(times(z0, z1)) -> a__times(mark(z0), mark(z1)) mark(square(z0)) -> a__square(mark(z0)) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(posrecip(z0)) -> posrecip(mark(z0)) mark(negrecip(z0)) -> negrecip(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(rnil) -> rnil mark(rcons(z0, z1)) -> rcons(mark(z0), mark(z1)) 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__FROM(z0) -> c(MARK(z0)) A__FROM(z0) -> c1 A__2NDSPOS(0, z0) -> c2 A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c3(MARK(z2)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c4(A__2NDSNEG(mark(z0), mark(z3)), MARK(z0)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c5(A__2NDSNEG(mark(z0), mark(z3)), MARK(z3)) A__2NDSPOS(z0, z1) -> c6 A__2NDSNEG(0, z0) -> c7 A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c8(MARK(z2)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c9(A__2NDSPOS(mark(z0), mark(z3)), MARK(z0)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c10(A__2NDSPOS(mark(z0), mark(z3)), MARK(z3)) A__2NDSNEG(z0, z1) -> c11 A__PI(z0) -> c12(A__2NDSPOS(mark(z0), a__from(0)), MARK(z0)) A__PI(z0) -> c13(A__2NDSPOS(mark(z0), a__from(0)), A__FROM(0)) A__PI(z0) -> c14 A__PLUS(0, z0) -> c15(MARK(z0)) A__PLUS(s(z0), z1) -> c16(A__PLUS(mark(z0), mark(z1)), MARK(z0)) A__PLUS(s(z0), z1) -> c17(A__PLUS(mark(z0), mark(z1)), MARK(z1)) A__PLUS(z0, z1) -> c18 A__TIMES(0, z0) -> c19 A__TIMES(s(z0), z1) -> c20(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), MARK(z1)) A__TIMES(s(z0), z1) -> c21(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z0)) A__TIMES(s(z0), z1) -> c22(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z1)) A__TIMES(z0, z1) -> c23 A__SQUARE(z0) -> c24(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c25(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c26 MARK(from(z0)) -> c27(A__FROM(mark(z0)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c28(A__2NDSPOS(mark(z0), mark(z1)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c29(A__2NDSPOS(mark(z0), mark(z1)), MARK(z1)) MARK(2ndsneg(z0, z1)) -> c30(A__2NDSNEG(mark(z0), mark(z1)), MARK(z0)) MARK(2ndsneg(z0, z1)) -> c31(A__2NDSNEG(mark(z0), mark(z1)), MARK(z1)) MARK(pi(z0)) -> c32(A__PI(mark(z0)), MARK(z0)) MARK(plus(z0, z1)) -> c33(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c34(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(times(z0, z1)) -> c35(A__TIMES(mark(z0), mark(z1)), MARK(z0)) MARK(times(z0, z1)) -> c36(A__TIMES(mark(z0), mark(z1)), MARK(z1)) MARK(square(z0)) -> c37(A__SQUARE(mark(z0)), MARK(z0)) MARK(0) -> c38 MARK(s(z0)) -> c39(MARK(z0)) MARK(posrecip(z0)) -> c40(MARK(z0)) MARK(negrecip(z0)) -> c41(MARK(z0)) MARK(nil) -> c42 MARK(cons(z0, z1)) -> c43(MARK(z0)) MARK(rnil) -> c44 MARK(rcons(z0, z1)) -> c45(MARK(z0)) MARK(rcons(z0, z1)) -> c46(MARK(z1)) The (relative) TRS S consists of the following rules: a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__2ndspos(0, z0) -> rnil a__2ndspos(s(z0), cons(z1, cons(z2, z3))) -> rcons(posrecip(mark(z2)), a__2ndsneg(mark(z0), mark(z3))) a__2ndspos(z0, z1) -> 2ndspos(z0, z1) a__2ndsneg(0, z0) -> rnil a__2ndsneg(s(z0), cons(z1, cons(z2, z3))) -> rcons(negrecip(mark(z2)), a__2ndspos(mark(z0), mark(z3))) a__2ndsneg(z0, z1) -> 2ndsneg(z0, z1) a__pi(z0) -> a__2ndspos(mark(z0), a__from(0)) a__pi(z0) -> pi(z0) a__plus(0, z0) -> mark(z0) a__plus(s(z0), z1) -> s(a__plus(mark(z0), mark(z1))) a__plus(z0, z1) -> plus(z0, z1) a__times(0, z0) -> 0 a__times(s(z0), z1) -> a__plus(mark(z1), a__times(mark(z0), mark(z1))) a__times(z0, z1) -> times(z0, z1) a__square(z0) -> a__times(mark(z0), mark(z0)) a__square(z0) -> square(z0) mark(from(z0)) -> a__from(mark(z0)) mark(2ndspos(z0, z1)) -> a__2ndspos(mark(z0), mark(z1)) mark(2ndsneg(z0, z1)) -> a__2ndsneg(mark(z0), mark(z1)) mark(pi(z0)) -> a__pi(mark(z0)) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(times(z0, z1)) -> a__times(mark(z0), mark(z1)) mark(square(z0)) -> a__square(mark(z0)) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(posrecip(z0)) -> posrecip(mark(z0)) mark(negrecip(z0)) -> negrecip(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(rnil) -> rnil mark(rcons(z0, z1)) -> rcons(mark(z0), mark(z1)) 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(from(z0)) ->^+ c27(A__FROM(mark(z0)), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / from(z0)]. 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__FROM(z0) -> c(MARK(z0)) A__FROM(z0) -> c1 A__2NDSPOS(0, z0) -> c2 A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c3(MARK(z2)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c4(A__2NDSNEG(mark(z0), mark(z3)), MARK(z0)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c5(A__2NDSNEG(mark(z0), mark(z3)), MARK(z3)) A__2NDSPOS(z0, z1) -> c6 A__2NDSNEG(0, z0) -> c7 A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c8(MARK(z2)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c9(A__2NDSPOS(mark(z0), mark(z3)), MARK(z0)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c10(A__2NDSPOS(mark(z0), mark(z3)), MARK(z3)) A__2NDSNEG(z0, z1) -> c11 A__PI(z0) -> c12(A__2NDSPOS(mark(z0), a__from(0)), MARK(z0)) A__PI(z0) -> c13(A__2NDSPOS(mark(z0), a__from(0)), A__FROM(0)) A__PI(z0) -> c14 A__PLUS(0, z0) -> c15(MARK(z0)) A__PLUS(s(z0), z1) -> c16(A__PLUS(mark(z0), mark(z1)), MARK(z0)) A__PLUS(s(z0), z1) -> c17(A__PLUS(mark(z0), mark(z1)), MARK(z1)) A__PLUS(z0, z1) -> c18 A__TIMES(0, z0) -> c19 A__TIMES(s(z0), z1) -> c20(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), MARK(z1)) A__TIMES(s(z0), z1) -> c21(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z0)) A__TIMES(s(z0), z1) -> c22(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z1)) A__TIMES(z0, z1) -> c23 A__SQUARE(z0) -> c24(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c25(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c26 MARK(from(z0)) -> c27(A__FROM(mark(z0)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c28(A__2NDSPOS(mark(z0), mark(z1)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c29(A__2NDSPOS(mark(z0), mark(z1)), MARK(z1)) MARK(2ndsneg(z0, z1)) -> c30(A__2NDSNEG(mark(z0), mark(z1)), MARK(z0)) MARK(2ndsneg(z0, z1)) -> c31(A__2NDSNEG(mark(z0), mark(z1)), MARK(z1)) MARK(pi(z0)) -> c32(A__PI(mark(z0)), MARK(z0)) MARK(plus(z0, z1)) -> c33(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c34(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(times(z0, z1)) -> c35(A__TIMES(mark(z0), mark(z1)), MARK(z0)) MARK(times(z0, z1)) -> c36(A__TIMES(mark(z0), mark(z1)), MARK(z1)) MARK(square(z0)) -> c37(A__SQUARE(mark(z0)), MARK(z0)) MARK(0) -> c38 MARK(s(z0)) -> c39(MARK(z0)) MARK(posrecip(z0)) -> c40(MARK(z0)) MARK(negrecip(z0)) -> c41(MARK(z0)) MARK(nil) -> c42 MARK(cons(z0, z1)) -> c43(MARK(z0)) MARK(rnil) -> c44 MARK(rcons(z0, z1)) -> c45(MARK(z0)) MARK(rcons(z0, z1)) -> c46(MARK(z1)) The (relative) TRS S consists of the following rules: a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__2ndspos(0, z0) -> rnil a__2ndspos(s(z0), cons(z1, cons(z2, z3))) -> rcons(posrecip(mark(z2)), a__2ndsneg(mark(z0), mark(z3))) a__2ndspos(z0, z1) -> 2ndspos(z0, z1) a__2ndsneg(0, z0) -> rnil a__2ndsneg(s(z0), cons(z1, cons(z2, z3))) -> rcons(negrecip(mark(z2)), a__2ndspos(mark(z0), mark(z3))) a__2ndsneg(z0, z1) -> 2ndsneg(z0, z1) a__pi(z0) -> a__2ndspos(mark(z0), a__from(0)) a__pi(z0) -> pi(z0) a__plus(0, z0) -> mark(z0) a__plus(s(z0), z1) -> s(a__plus(mark(z0), mark(z1))) a__plus(z0, z1) -> plus(z0, z1) a__times(0, z0) -> 0 a__times(s(z0), z1) -> a__plus(mark(z1), a__times(mark(z0), mark(z1))) a__times(z0, z1) -> times(z0, z1) a__square(z0) -> a__times(mark(z0), mark(z0)) a__square(z0) -> square(z0) mark(from(z0)) -> a__from(mark(z0)) mark(2ndspos(z0, z1)) -> a__2ndspos(mark(z0), mark(z1)) mark(2ndsneg(z0, z1)) -> a__2ndsneg(mark(z0), mark(z1)) mark(pi(z0)) -> a__pi(mark(z0)) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(times(z0, z1)) -> a__times(mark(z0), mark(z1)) mark(square(z0)) -> a__square(mark(z0)) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(posrecip(z0)) -> posrecip(mark(z0)) mark(negrecip(z0)) -> negrecip(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(rnil) -> rnil mark(rcons(z0, z1)) -> rcons(mark(z0), mark(z1)) 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__FROM(z0) -> c(MARK(z0)) A__FROM(z0) -> c1 A__2NDSPOS(0, z0) -> c2 A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c3(MARK(z2)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c4(A__2NDSNEG(mark(z0), mark(z3)), MARK(z0)) A__2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> c5(A__2NDSNEG(mark(z0), mark(z3)), MARK(z3)) A__2NDSPOS(z0, z1) -> c6 A__2NDSNEG(0, z0) -> c7 A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c8(MARK(z2)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c9(A__2NDSPOS(mark(z0), mark(z3)), MARK(z0)) A__2NDSNEG(s(z0), cons(z1, cons(z2, z3))) -> c10(A__2NDSPOS(mark(z0), mark(z3)), MARK(z3)) A__2NDSNEG(z0, z1) -> c11 A__PI(z0) -> c12(A__2NDSPOS(mark(z0), a__from(0)), MARK(z0)) A__PI(z0) -> c13(A__2NDSPOS(mark(z0), a__from(0)), A__FROM(0)) A__PI(z0) -> c14 A__PLUS(0, z0) -> c15(MARK(z0)) A__PLUS(s(z0), z1) -> c16(A__PLUS(mark(z0), mark(z1)), MARK(z0)) A__PLUS(s(z0), z1) -> c17(A__PLUS(mark(z0), mark(z1)), MARK(z1)) A__PLUS(z0, z1) -> c18 A__TIMES(0, z0) -> c19 A__TIMES(s(z0), z1) -> c20(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), MARK(z1)) A__TIMES(s(z0), z1) -> c21(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z0)) A__TIMES(s(z0), z1) -> c22(A__PLUS(mark(z1), a__times(mark(z0), mark(z1))), A__TIMES(mark(z0), mark(z1)), MARK(z1)) A__TIMES(z0, z1) -> c23 A__SQUARE(z0) -> c24(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c25(A__TIMES(mark(z0), mark(z0)), MARK(z0)) A__SQUARE(z0) -> c26 MARK(from(z0)) -> c27(A__FROM(mark(z0)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c28(A__2NDSPOS(mark(z0), mark(z1)), MARK(z0)) MARK(2ndspos(z0, z1)) -> c29(A__2NDSPOS(mark(z0), mark(z1)), MARK(z1)) MARK(2ndsneg(z0, z1)) -> c30(A__2NDSNEG(mark(z0), mark(z1)), MARK(z0)) MARK(2ndsneg(z0, z1)) -> c31(A__2NDSNEG(mark(z0), mark(z1)), MARK(z1)) MARK(pi(z0)) -> c32(A__PI(mark(z0)), MARK(z0)) MARK(plus(z0, z1)) -> c33(A__PLUS(mark(z0), mark(z1)), MARK(z0)) MARK(plus(z0, z1)) -> c34(A__PLUS(mark(z0), mark(z1)), MARK(z1)) MARK(times(z0, z1)) -> c35(A__TIMES(mark(z0), mark(z1)), MARK(z0)) MARK(times(z0, z1)) -> c36(A__TIMES(mark(z0), mark(z1)), MARK(z1)) MARK(square(z0)) -> c37(A__SQUARE(mark(z0)), MARK(z0)) MARK(0) -> c38 MARK(s(z0)) -> c39(MARK(z0)) MARK(posrecip(z0)) -> c40(MARK(z0)) MARK(negrecip(z0)) -> c41(MARK(z0)) MARK(nil) -> c42 MARK(cons(z0, z1)) -> c43(MARK(z0)) MARK(rnil) -> c44 MARK(rcons(z0, z1)) -> c45(MARK(z0)) MARK(rcons(z0, z1)) -> c46(MARK(z1)) The (relative) TRS S consists of the following rules: a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__2ndspos(0, z0) -> rnil a__2ndspos(s(z0), cons(z1, cons(z2, z3))) -> rcons(posrecip(mark(z2)), a__2ndsneg(mark(z0), mark(z3))) a__2ndspos(z0, z1) -> 2ndspos(z0, z1) a__2ndsneg(0, z0) -> rnil a__2ndsneg(s(z0), cons(z1, cons(z2, z3))) -> rcons(negrecip(mark(z2)), a__2ndspos(mark(z0), mark(z3))) a__2ndsneg(z0, z1) -> 2ndsneg(z0, z1) a__pi(z0) -> a__2ndspos(mark(z0), a__from(0)) a__pi(z0) -> pi(z0) a__plus(0, z0) -> mark(z0) a__plus(s(z0), z1) -> s(a__plus(mark(z0), mark(z1))) a__plus(z0, z1) -> plus(z0, z1) a__times(0, z0) -> 0 a__times(s(z0), z1) -> a__plus(mark(z1), a__times(mark(z0), mark(z1))) a__times(z0, z1) -> times(z0, z1) a__square(z0) -> a__times(mark(z0), mark(z0)) a__square(z0) -> square(z0) mark(from(z0)) -> a__from(mark(z0)) mark(2ndspos(z0, z1)) -> a__2ndspos(mark(z0), mark(z1)) mark(2ndsneg(z0, z1)) -> a__2ndsneg(mark(z0), mark(z1)) mark(pi(z0)) -> a__pi(mark(z0)) mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1)) mark(times(z0, z1)) -> a__times(mark(z0), mark(z1)) mark(square(z0)) -> a__square(mark(z0)) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(posrecip(z0)) -> posrecip(mark(z0)) mark(negrecip(z0)) -> negrecip(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(rnil) -> rnil mark(rcons(z0, z1)) -> rcons(mark(z0), mark(z1)) Rewrite Strategy: INNERMOST