/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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), 662 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, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) U15(tt, V2) -> U16(isNat(activate(V2))) U16(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V2) -> U32(isNatKind(activate(V2))) U32(tt) -> tt U41(tt) -> tt U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) U52(tt, N) -> activate(N) U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) U64(tt, M, N) -> s(plus(activate(N), activate(M))) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N, 0) -> U51(isNat(N), N) plus(N, s(M)) -> U61(isNat(M), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) 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, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) U15(tt, V2) -> U16(isNat(activate(V2))) U16(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V2) -> U32(isNatKind(activate(V2))) U32(tt) -> tt U41(tt) -> tt U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) U52(tt, N) -> activate(N) U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) U64(tt, M, N) -> s(plus(activate(N), activate(M))) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N, 0') -> U51(isNat(N), N) plus(N, s(M)) -> U61(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) U15(tt, V2) -> U16(isNat(activate(V2))) U16(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V2) -> U32(isNatKind(activate(V2))) U32(tt) -> tt U41(tt) -> tt U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) U52(tt, N) -> activate(N) U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) U64(tt, M, N) -> s(plus(activate(N), activate(M))) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N, 0') -> U51(isNat(N), N) plus(N, s(M)) -> U61(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt tt :: tt U12 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt isNatKind :: n__0:n__plus:n__s -> tt activate :: n__0:n__plus:n__s -> n__0:n__plus:n__s U13 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U14 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U15 :: tt -> n__0:n__plus:n__s -> tt isNat :: n__0:n__plus:n__s -> tt U16 :: tt -> tt U21 :: tt -> n__0:n__plus:n__s -> tt U22 :: tt -> n__0:n__plus:n__s -> tt U23 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s -> tt U32 :: tt -> tt U41 :: tt -> tt U51 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U52 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U61 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U62 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U63 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U64 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s s :: n__0:n__plus:n__s -> n__0:n__plus:n__s plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__0 :: n__0:n__plus:n__s n__plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__s :: n__0:n__plus:n__s -> n__0:n__plus:n__s 0' :: n__0:n__plus:n__s hole_tt1_7 :: tt hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s gen_n__0:n__plus:n__s3_7 :: Nat -> n__0:n__plus:n__s ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: isNatKind, activate, isNat, U51, U52, plus They will be analysed ascendingly in the following order: isNatKind = activate isNatKind = isNat isNatKind = U51 isNatKind = U52 isNatKind = plus activate = isNat activate = U51 activate = U52 activate = plus isNat = U51 isNat = U52 isNat = plus U51 = U52 U51 = plus U52 = plus ---------------------------------------- (6) Obligation: TRS: Rules: U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) U15(tt, V2) -> U16(isNat(activate(V2))) U16(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V2) -> U32(isNatKind(activate(V2))) U32(tt) -> tt U41(tt) -> tt U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) U52(tt, N) -> activate(N) U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) U64(tt, M, N) -> s(plus(activate(N), activate(M))) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N, 0') -> U51(isNat(N), N) plus(N, s(M)) -> U61(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt tt :: tt U12 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt isNatKind :: n__0:n__plus:n__s -> tt activate :: n__0:n__plus:n__s -> n__0:n__plus:n__s U13 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U14 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U15 :: tt -> n__0:n__plus:n__s -> tt isNat :: n__0:n__plus:n__s -> tt U16 :: tt -> tt U21 :: tt -> n__0:n__plus:n__s -> tt U22 :: tt -> n__0:n__plus:n__s -> tt U23 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s -> tt U32 :: tt -> tt U41 :: tt -> tt U51 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U52 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U61 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U62 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U63 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U64 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s s :: n__0:n__plus:n__s -> n__0:n__plus:n__s plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__0 :: n__0:n__plus:n__s n__plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__s :: n__0:n__plus:n__s -> n__0:n__plus:n__s 0' :: n__0:n__plus:n__s hole_tt1_7 :: tt hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s gen_n__0:n__plus:n__s3_7 :: Nat -> n__0:n__plus:n__s Generator Equations: gen_n__0:n__plus:n__s3_7(0) <=> n__0 gen_n__0:n__plus:n__s3_7(+(x, 1)) <=> n__plus(gen_n__0:n__plus:n__s3_7(x), n__0) The following defined symbols remain to be analysed: activate, isNatKind, isNat, U51, U52, plus They will be analysed ascendingly in the following order: isNatKind = activate isNatKind = isNat isNatKind = U51 isNatKind = U52 isNatKind = plus activate = isNat activate = U51 activate = U52 activate = plus isNat = U51 isNat = U52 isNat = plus U51 = U52 U51 = plus U52 = plus ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: isNatKind(gen_n__0:n__plus:n__s3_7(n5934_7)) -> tt, rt in Omega(1 + n5934_7) Induction Base: isNatKind(gen_n__0:n__plus:n__s3_7(0)) ->_R^Omega(1) tt Induction Step: isNatKind(gen_n__0:n__plus:n__s3_7(+(n5934_7, 1))) ->_R^Omega(1) U31(isNatKind(activate(gen_n__0:n__plus:n__s3_7(n5934_7))), activate(n__0)) ->_R^Omega(1) U31(isNatKind(gen_n__0:n__plus:n__s3_7(n5934_7)), activate(n__0)) ->_IH U31(tt, activate(n__0)) ->_R^Omega(1) U31(tt, n__0) ->_R^Omega(1) U32(isNatKind(activate(n__0))) ->_R^Omega(1) U32(isNatKind(n__0)) ->_R^Omega(1) U32(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, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) U15(tt, V2) -> U16(isNat(activate(V2))) U16(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V2) -> U32(isNatKind(activate(V2))) U32(tt) -> tt U41(tt) -> tt U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) U52(tt, N) -> activate(N) U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) U64(tt, M, N) -> s(plus(activate(N), activate(M))) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N, 0') -> U51(isNat(N), N) plus(N, s(M)) -> U61(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt tt :: tt U12 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt isNatKind :: n__0:n__plus:n__s -> tt activate :: n__0:n__plus:n__s -> n__0:n__plus:n__s U13 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U14 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U15 :: tt -> n__0:n__plus:n__s -> tt isNat :: n__0:n__plus:n__s -> tt U16 :: tt -> tt U21 :: tt -> n__0:n__plus:n__s -> tt U22 :: tt -> n__0:n__plus:n__s -> tt U23 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s -> tt U32 :: tt -> tt U41 :: tt -> tt U51 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U52 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U61 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U62 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U63 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U64 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s s :: n__0:n__plus:n__s -> n__0:n__plus:n__s plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__0 :: n__0:n__plus:n__s n__plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__s :: n__0:n__plus:n__s -> n__0:n__plus:n__s 0' :: n__0:n__plus:n__s hole_tt1_7 :: tt hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s gen_n__0:n__plus:n__s3_7 :: Nat -> n__0:n__plus:n__s Generator Equations: gen_n__0:n__plus:n__s3_7(0) <=> n__0 gen_n__0:n__plus:n__s3_7(+(x, 1)) <=> n__plus(gen_n__0:n__plus:n__s3_7(x), n__0) The following defined symbols remain to be analysed: isNatKind, isNat They will be analysed ascendingly in the following order: isNatKind = activate isNatKind = isNat isNatKind = U51 isNatKind = U52 isNatKind = plus activate = isNat activate = U51 activate = U52 activate = plus isNat = U51 isNat = U52 isNat = plus U51 = U52 U51 = plus U52 = plus ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: U11(tt, V1, V2) -> U12(isNatKind(activate(V1)), activate(V1), activate(V2)) U12(tt, V1, V2) -> U13(isNatKind(activate(V2)), activate(V1), activate(V2)) U13(tt, V1, V2) -> U14(isNatKind(activate(V2)), activate(V1), activate(V2)) U14(tt, V1, V2) -> U15(isNat(activate(V1)), activate(V2)) U15(tt, V2) -> U16(isNat(activate(V2))) U16(tt) -> tt U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) U22(tt, V1) -> U23(isNat(activate(V1))) U23(tt) -> tt U31(tt, V2) -> U32(isNatKind(activate(V2))) U32(tt) -> tt U41(tt) -> tt U51(tt, N) -> U52(isNatKind(activate(N)), activate(N)) U52(tt, N) -> activate(N) U61(tt, M, N) -> U62(isNatKind(activate(M)), activate(M), activate(N)) U62(tt, M, N) -> U63(isNat(activate(N)), activate(M), activate(N)) U63(tt, M, N) -> U64(isNatKind(activate(N)), activate(M), activate(N)) U64(tt, M, N) -> s(plus(activate(N), activate(M))) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(isNatKind(activate(V1)), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> U31(isNatKind(activate(V1)), activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N, 0') -> U51(isNat(N), N) plus(N, s(M)) -> U61(isNat(M), M, N) 0' -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) activate(n__0) -> 0' activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__s(X)) -> s(X) activate(X) -> X Types: U11 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt tt :: tt U12 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt isNatKind :: n__0:n__plus:n__s -> tt activate :: n__0:n__plus:n__s -> n__0:n__plus:n__s U13 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U14 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> tt U15 :: tt -> n__0:n__plus:n__s -> tt isNat :: n__0:n__plus:n__s -> tt U16 :: tt -> tt U21 :: tt -> n__0:n__plus:n__s -> tt U22 :: tt -> n__0:n__plus:n__s -> tt U23 :: tt -> tt U31 :: tt -> n__0:n__plus:n__s -> tt U32 :: tt -> tt U41 :: tt -> tt U51 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U52 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s U61 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U62 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U63 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s U64 :: tt -> n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s s :: n__0:n__plus:n__s -> n__0:n__plus:n__s plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__0 :: n__0:n__plus:n__s n__plus :: n__0:n__plus:n__s -> n__0:n__plus:n__s -> n__0:n__plus:n__s n__s :: n__0:n__plus:n__s -> n__0:n__plus:n__s 0' :: n__0:n__plus:n__s hole_tt1_7 :: tt hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s gen_n__0:n__plus:n__s3_7 :: Nat -> n__0:n__plus:n__s Lemmas: isNatKind(gen_n__0:n__plus:n__s3_7(n5934_7)) -> tt, rt in Omega(1 + n5934_7) Generator Equations: gen_n__0:n__plus:n__s3_7(0) <=> n__0 gen_n__0:n__plus:n__s3_7(+(x, 1)) <=> n__plus(gen_n__0:n__plus:n__s3_7(x), n__0) The following defined symbols remain to be analysed: isNat, activate, U51, U52, plus They will be analysed ascendingly in the following order: isNatKind = activate isNatKind = isNat isNatKind = U51 isNatKind = U52 isNatKind = plus activate = isNat activate = U51 activate = U52 activate = plus isNat = U51 isNat = U52 isNat = plus U51 = U52 U51 = plus U52 = plus