WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 0 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 365 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: U11(tt, V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt, V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) U52(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) plus(N, 0) -> U41(isNat(N), N) plus(N, s(M)) -> U51(isNat(M), M, N) x(N, 0) -> U61(isNat(N)) x(N, s(M)) -> U71(isNat(M), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(n__x(X1, X2)) -> x(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: U11(tt, V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt, V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) U52(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0' U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) plus(N, 0') -> U41(isNat(N), N) plus(N, s(M)) -> U51(isNat(M), M, N) x(N, 0') -> U61(isNat(N)) x(N, s(M)) -> U71(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(n__x(X1, X2)) -> x(X1, X2) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: U11(tt, V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt, V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) U52(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0' U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) plus(N, 0') -> U41(isNat(N), N) plus(N, s(M)) -> U51(isNat(M), M, N) x(N, 0') -> U61(isNat(N)) x(N, s(M)) -> U71(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(n__x(X1, X2)) -> x(X1, X2) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s:n__x -> tt tt :: tt U12 :: tt -> tt isNat :: n__0:n__plus:n__s:n__x -> tt activate :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U21 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s:n__x -> tt U32 :: tt -> tt U41 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U51 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U52 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U61 :: tt -> n__0:n__plus:n__s:n__x 0' :: n__0:n__plus:n__s:n__x U71 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U72 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__0 :: n__0:n__plus:n__s:n__x n__plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x hole_tt1_3 :: tt hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x gen_n__0:n__plus:n__s:n__x3_3 :: Nat -> n__0:n__plus:n__s:n__x ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: isNat, activate, U41, plus, x They will be analysed ascendingly in the following order: isNat = activate isNat = U41 isNat = plus isNat = x activate = U41 activate = plus activate = x U41 = plus U41 = x plus = x ---------------------------------------- (6) Obligation: TRS: Rules: U11(tt, V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt, V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) U52(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0' U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) plus(N, 0') -> U41(isNat(N), N) plus(N, s(M)) -> U51(isNat(M), M, N) x(N, 0') -> U61(isNat(N)) x(N, s(M)) -> U71(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(n__x(X1, X2)) -> x(X1, X2) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s:n__x -> tt tt :: tt U12 :: tt -> tt isNat :: n__0:n__plus:n__s:n__x -> tt activate :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U21 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s:n__x -> tt U32 :: tt -> tt U41 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U51 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U52 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U61 :: tt -> n__0:n__plus:n__s:n__x 0' :: n__0:n__plus:n__s:n__x U71 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U72 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__0 :: n__0:n__plus:n__s:n__x n__plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x hole_tt1_3 :: tt hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x gen_n__0:n__plus:n__s:n__x3_3 :: Nat -> n__0:n__plus:n__s:n__x Generator Equations: gen_n__0:n__plus:n__s:n__x3_3(0) <=> n__0 gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) <=> n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0) The following defined symbols remain to be analysed: activate, isNat, U41, plus, x They will be analysed ascendingly in the following order: isNat = activate isNat = U41 isNat = plus isNat = x activate = U41 activate = plus activate = x U41 = plus U41 = x plus = x ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) -> tt, rt in Omega(1 + n225_3) Induction Base: isNat(gen_n__0:n__plus:n__s:n__x3_3(0)) ->_R^Omega(1) tt Induction Step: isNat(gen_n__0:n__plus:n__s:n__x3_3(+(n225_3, 1))) ->_R^Omega(1) U11(isNat(activate(gen_n__0:n__plus:n__s:n__x3_3(n225_3))), activate(n__0)) ->_R^Omega(1) U11(isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)), activate(n__0)) ->_IH U11(tt, activate(n__0)) ->_R^Omega(1) U11(tt, n__0) ->_R^Omega(1) U12(isNat(activate(n__0))) ->_R^Omega(1) U12(isNat(n__0)) ->_R^Omega(1) U12(tt) ->_R^Omega(1) tt We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: U11(tt, V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt, V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) U52(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0' U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) plus(N, 0') -> U41(isNat(N), N) plus(N, s(M)) -> U51(isNat(M), M, N) x(N, 0') -> U61(isNat(N)) x(N, s(M)) -> U71(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(n__x(X1, X2)) -> x(X1, X2) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s:n__x -> tt tt :: tt U12 :: tt -> tt isNat :: n__0:n__plus:n__s:n__x -> tt activate :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U21 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s:n__x -> tt U32 :: tt -> tt U41 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U51 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U52 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U61 :: tt -> n__0:n__plus:n__s:n__x 0' :: n__0:n__plus:n__s:n__x U71 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U72 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__0 :: n__0:n__plus:n__s:n__x n__plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x hole_tt1_3 :: tt hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x gen_n__0:n__plus:n__s:n__x3_3 :: Nat -> n__0:n__plus:n__s:n__x Generator Equations: gen_n__0:n__plus:n__s:n__x3_3(0) <=> n__0 gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) <=> n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0) The following defined symbols remain to be analysed: isNat, x They will be analysed ascendingly in the following order: isNat = activate isNat = U41 isNat = plus isNat = x activate = U41 activate = plus activate = x U41 = plus U41 = x plus = x ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: U11(tt, V2) -> U12(isNat(activate(V2))) U12(tt) -> tt U21(tt) -> tt U31(tt, V2) -> U32(isNat(activate(V2))) U32(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> U52(isNat(activate(N)), activate(M), activate(N)) U52(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0' U71(tt, M, N) -> U72(isNat(activate(N)), activate(M), activate(N)) U72(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) isNat(n__x(V1, V2)) -> U31(isNat(activate(V1)), activate(V2)) plus(N, 0') -> U41(isNat(N), N) plus(N, s(M)) -> U51(isNat(M), M, N) x(N, 0') -> U61(isNat(N)) x(N, s(M)) -> U71(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(n__x(X1, X2)) -> x(X1, X2) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s:n__x -> tt tt :: tt U12 :: tt -> tt isNat :: n__0:n__plus:n__s:n__x -> tt activate :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U21 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s:n__x -> tt U32 :: tt -> tt U41 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U51 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U52 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U61 :: tt -> n__0:n__plus:n__s:n__x 0' :: n__0:n__plus:n__s:n__x U71 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x U72 :: tt -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__0 :: n__0:n__plus:n__s:n__x n__plus :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__s :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x n__x :: n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x -> n__0:n__plus:n__s:n__x hole_tt1_3 :: tt hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x gen_n__0:n__plus:n__s:n__x3_3 :: Nat -> n__0:n__plus:n__s:n__x Lemmas: isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) -> tt, rt in Omega(1 + n225_3) Generator Equations: gen_n__0:n__plus:n__s:n__x3_3(0) <=> n__0 gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) <=> n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0) The following defined symbols remain to be analysed: x, activate, U41, plus They will be analysed ascendingly in the following order: isNat = activate isNat = U41 isNat = plus isNat = x activate = U41 activate = plus activate = x U41 = plus U41 = x plus = x