12.86/4.08 WORST_CASE(NON_POLY, ?) 12.86/4.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 12.86/4.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.86/4.08 12.86/4.08 12.86/4.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 12.86/4.08 12.86/4.08 (0) CpxTRS 12.86/4.08 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 12.86/4.08 (2) TRS for Loop Detection 12.86/4.08 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 12.86/4.08 (4) BEST 12.86/4.08 (5) proven lower bound 12.86/4.08 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 12.86/4.08 (7) BOUNDS(n^1, INF) 12.86/4.08 (8) TRS for Loop Detection 12.86/4.08 (9) DecreasingLoopProof [FINISHED, 1847 ms] 12.86/4.08 (10) BOUNDS(EXP, INF) 12.86/4.08 12.86/4.08 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (0) 12.86/4.08 Obligation: 12.86/4.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 12.86/4.08 12.86/4.08 12.86/4.08 The TRS R consists of the following rules: 12.86/4.08 12.86/4.08 a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) 12.86/4.08 a__U12(tt, V2) -> a__U13(a__isNat(V2)) 12.86/4.08 a__U13(tt) -> tt 12.86/4.08 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 12.86/4.08 a__U22(tt) -> tt 12.86/4.08 a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) 12.86/4.08 a__U32(tt, V2) -> a__U33(a__isNat(V2)) 12.86/4.08 a__U33(tt) -> tt 12.86/4.08 a__U41(tt, N) -> mark(N) 12.86/4.08 a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) 12.86/4.08 a__U61(tt) -> 0 12.86/4.08 a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 12.86/4.08 a__and(tt, X) -> mark(X) 12.86/4.08 a__isNat(0) -> tt 12.86/4.08 a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 12.86/4.08 a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNatKind(0) -> tt 12.86/4.08 a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__isNatKind(s(V1)) -> a__isNatKind(V1) 12.86/4.08 a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 12.86/4.08 a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 12.86/4.08 a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 12.86/4.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 12.86/4.08 mark(isNat(X)) -> a__isNat(X) 12.86/4.08 mark(U13(X)) -> a__U13(mark(X)) 12.86/4.08 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 12.86/4.08 mark(U22(X)) -> a__U22(mark(X)) 12.86/4.08 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 12.86/4.08 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 12.86/4.08 mark(U33(X)) -> a__U33(mark(X)) 12.86/4.08 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 12.86/4.08 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 12.86/4.08 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 12.86/4.08 mark(U61(X)) -> a__U61(mark(X)) 12.86/4.08 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 12.86/4.08 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 12.86/4.08 mark(and(X1, X2)) -> a__and(mark(X1), X2) 12.86/4.08 mark(isNatKind(X)) -> a__isNatKind(X) 12.86/4.08 mark(tt) -> tt 12.86/4.08 mark(s(X)) -> s(mark(X)) 12.86/4.08 mark(0) -> 0 12.86/4.08 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 12.86/4.08 a__U12(X1, X2) -> U12(X1, X2) 12.86/4.08 a__isNat(X) -> isNat(X) 12.86/4.08 a__U13(X) -> U13(X) 12.86/4.08 a__U21(X1, X2) -> U21(X1, X2) 12.86/4.08 a__U22(X) -> U22(X) 12.86/4.08 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 12.86/4.08 a__U32(X1, X2) -> U32(X1, X2) 12.86/4.08 a__U33(X) -> U33(X) 12.86/4.08 a__U41(X1, X2) -> U41(X1, X2) 12.86/4.08 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 12.86/4.08 a__plus(X1, X2) -> plus(X1, X2) 12.86/4.08 a__U61(X) -> U61(X) 12.86/4.08 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 12.86/4.08 a__x(X1, X2) -> x(X1, X2) 12.86/4.08 a__and(X1, X2) -> and(X1, X2) 12.86/4.08 a__isNatKind(X) -> isNatKind(X) 12.86/4.08 12.86/4.08 S is empty. 12.86/4.08 Rewrite Strategy: FULL 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 12.86/4.08 Transformed a relative TRS into a decreasing-loop problem. 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (2) 12.86/4.08 Obligation: 12.86/4.08 Analyzing the following TRS for decreasing loops: 12.86/4.08 12.86/4.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 12.86/4.08 12.86/4.08 12.86/4.08 The TRS R consists of the following rules: 12.86/4.08 12.86/4.08 a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) 12.86/4.08 a__U12(tt, V2) -> a__U13(a__isNat(V2)) 12.86/4.08 a__U13(tt) -> tt 12.86/4.08 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 12.86/4.08 a__U22(tt) -> tt 12.86/4.08 a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) 12.86/4.08 a__U32(tt, V2) -> a__U33(a__isNat(V2)) 12.86/4.08 a__U33(tt) -> tt 12.86/4.08 a__U41(tt, N) -> mark(N) 12.86/4.08 a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) 12.86/4.08 a__U61(tt) -> 0 12.86/4.08 a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 12.86/4.08 a__and(tt, X) -> mark(X) 12.86/4.08 a__isNat(0) -> tt 12.86/4.08 a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 12.86/4.08 a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNatKind(0) -> tt 12.86/4.08 a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__isNatKind(s(V1)) -> a__isNatKind(V1) 12.86/4.08 a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 12.86/4.08 a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 12.86/4.08 a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 12.86/4.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 12.86/4.08 mark(isNat(X)) -> a__isNat(X) 12.86/4.08 mark(U13(X)) -> a__U13(mark(X)) 12.86/4.08 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 12.86/4.08 mark(U22(X)) -> a__U22(mark(X)) 12.86/4.08 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 12.86/4.08 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 12.86/4.08 mark(U33(X)) -> a__U33(mark(X)) 12.86/4.08 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 12.86/4.08 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 12.86/4.08 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 12.86/4.08 mark(U61(X)) -> a__U61(mark(X)) 12.86/4.08 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 12.86/4.08 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 12.86/4.08 mark(and(X1, X2)) -> a__and(mark(X1), X2) 12.86/4.08 mark(isNatKind(X)) -> a__isNatKind(X) 12.86/4.08 mark(tt) -> tt 12.86/4.08 mark(s(X)) -> s(mark(X)) 12.86/4.08 mark(0) -> 0 12.86/4.08 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 12.86/4.08 a__U12(X1, X2) -> U12(X1, X2) 12.86/4.08 a__isNat(X) -> isNat(X) 12.86/4.08 a__U13(X) -> U13(X) 12.86/4.08 a__U21(X1, X2) -> U21(X1, X2) 12.86/4.08 a__U22(X) -> U22(X) 12.86/4.08 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 12.86/4.08 a__U32(X1, X2) -> U32(X1, X2) 12.86/4.08 a__U33(X) -> U33(X) 12.86/4.08 a__U41(X1, X2) -> U41(X1, X2) 12.86/4.08 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 12.86/4.08 a__plus(X1, X2) -> plus(X1, X2) 12.86/4.08 a__U61(X) -> U61(X) 12.86/4.08 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 12.86/4.08 a__x(X1, X2) -> x(X1, X2) 12.86/4.08 a__and(X1, X2) -> and(X1, X2) 12.86/4.08 a__isNatKind(X) -> isNatKind(X) 12.86/4.08 12.86/4.08 S is empty. 12.86/4.08 Rewrite Strategy: FULL 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (3) DecreasingLoopProof (LOWER BOUND(ID)) 12.86/4.08 The following loop(s) give(s) rise to the lower bound Omega(n^1): 12.86/4.08 12.86/4.08 The rewrite sequence 12.86/4.08 12.86/4.08 mark(U41(X1, X2)) ->^+ a__U41(mark(X1), X2) 12.86/4.08 12.86/4.08 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 12.86/4.08 12.86/4.08 The pumping substitution is [X1 / U41(X1, X2)]. 12.86/4.08 12.86/4.08 The result substitution is [ ]. 12.86/4.08 12.86/4.08 12.86/4.08 12.86/4.08 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (4) 12.86/4.08 Complex Obligation (BEST) 12.86/4.08 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (5) 12.86/4.08 Obligation: 12.86/4.08 Proved the lower bound n^1 for the following obligation: 12.86/4.08 12.86/4.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 12.86/4.08 12.86/4.08 12.86/4.08 The TRS R consists of the following rules: 12.86/4.08 12.86/4.08 a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) 12.86/4.08 a__U12(tt, V2) -> a__U13(a__isNat(V2)) 12.86/4.08 a__U13(tt) -> tt 12.86/4.08 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 12.86/4.08 a__U22(tt) -> tt 12.86/4.08 a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) 12.86/4.08 a__U32(tt, V2) -> a__U33(a__isNat(V2)) 12.86/4.08 a__U33(tt) -> tt 12.86/4.08 a__U41(tt, N) -> mark(N) 12.86/4.08 a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) 12.86/4.08 a__U61(tt) -> 0 12.86/4.08 a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 12.86/4.08 a__and(tt, X) -> mark(X) 12.86/4.08 a__isNat(0) -> tt 12.86/4.08 a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 12.86/4.08 a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNatKind(0) -> tt 12.86/4.08 a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__isNatKind(s(V1)) -> a__isNatKind(V1) 12.86/4.08 a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 12.86/4.08 a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 12.86/4.08 a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 12.86/4.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 12.86/4.08 mark(isNat(X)) -> a__isNat(X) 12.86/4.08 mark(U13(X)) -> a__U13(mark(X)) 12.86/4.08 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 12.86/4.08 mark(U22(X)) -> a__U22(mark(X)) 12.86/4.08 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 12.86/4.08 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 12.86/4.08 mark(U33(X)) -> a__U33(mark(X)) 12.86/4.08 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 12.86/4.08 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 12.86/4.08 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 12.86/4.08 mark(U61(X)) -> a__U61(mark(X)) 12.86/4.08 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 12.86/4.08 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 12.86/4.08 mark(and(X1, X2)) -> a__and(mark(X1), X2) 12.86/4.08 mark(isNatKind(X)) -> a__isNatKind(X) 12.86/4.08 mark(tt) -> tt 12.86/4.08 mark(s(X)) -> s(mark(X)) 12.86/4.08 mark(0) -> 0 12.86/4.08 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 12.86/4.08 a__U12(X1, X2) -> U12(X1, X2) 12.86/4.08 a__isNat(X) -> isNat(X) 12.86/4.08 a__U13(X) -> U13(X) 12.86/4.08 a__U21(X1, X2) -> U21(X1, X2) 12.86/4.08 a__U22(X) -> U22(X) 12.86/4.08 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 12.86/4.08 a__U32(X1, X2) -> U32(X1, X2) 12.86/4.08 a__U33(X) -> U33(X) 12.86/4.08 a__U41(X1, X2) -> U41(X1, X2) 12.86/4.08 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 12.86/4.08 a__plus(X1, X2) -> plus(X1, X2) 12.86/4.08 a__U61(X) -> U61(X) 12.86/4.08 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 12.86/4.08 a__x(X1, X2) -> x(X1, X2) 12.86/4.08 a__and(X1, X2) -> and(X1, X2) 12.86/4.08 a__isNatKind(X) -> isNatKind(X) 12.86/4.08 12.86/4.08 S is empty. 12.86/4.08 Rewrite Strategy: FULL 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (6) LowerBoundPropagationProof (FINISHED) 12.86/4.08 Propagated lower bound. 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (7) 12.86/4.08 BOUNDS(n^1, INF) 12.86/4.08 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (8) 12.86/4.08 Obligation: 12.86/4.08 Analyzing the following TRS for decreasing loops: 12.86/4.08 12.86/4.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 12.86/4.08 12.86/4.08 12.86/4.08 The TRS R consists of the following rules: 12.86/4.08 12.86/4.08 a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2) 12.86/4.08 a__U12(tt, V2) -> a__U13(a__isNat(V2)) 12.86/4.08 a__U13(tt) -> tt 12.86/4.08 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 12.86/4.08 a__U22(tt) -> tt 12.86/4.08 a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2) 12.86/4.08 a__U32(tt, V2) -> a__U33(a__isNat(V2)) 12.86/4.08 a__U33(tt) -> tt 12.86/4.08 a__U41(tt, N) -> mark(N) 12.86/4.08 a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M))) 12.86/4.08 a__U61(tt) -> 0 12.86/4.08 a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) 12.86/4.08 a__and(tt, X) -> mark(X) 12.86/4.08 a__isNat(0) -> tt 12.86/4.08 a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 12.86/4.08 a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 12.86/4.08 a__isNatKind(0) -> tt 12.86/4.08 a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__isNatKind(s(V1)) -> a__isNatKind(V1) 12.86/4.08 a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2)) 12.86/4.08 a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N) 12.86/4.08 a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N))) 12.86/4.08 a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 12.86/4.08 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 12.86/4.08 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 12.86/4.08 mark(isNat(X)) -> a__isNat(X) 12.86/4.08 mark(U13(X)) -> a__U13(mark(X)) 12.86/4.08 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 12.86/4.08 mark(U22(X)) -> a__U22(mark(X)) 12.86/4.08 mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) 12.86/4.08 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 12.86/4.08 mark(U33(X)) -> a__U33(mark(X)) 12.86/4.08 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 12.86/4.08 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 12.86/4.08 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 12.86/4.08 mark(U61(X)) -> a__U61(mark(X)) 12.86/4.08 mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3) 12.86/4.08 mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) 12.86/4.08 mark(and(X1, X2)) -> a__and(mark(X1), X2) 12.86/4.08 mark(isNatKind(X)) -> a__isNatKind(X) 12.86/4.08 mark(tt) -> tt 12.86/4.08 mark(s(X)) -> s(mark(X)) 12.86/4.08 mark(0) -> 0 12.86/4.08 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 12.86/4.08 a__U12(X1, X2) -> U12(X1, X2) 12.86/4.08 a__isNat(X) -> isNat(X) 12.86/4.08 a__U13(X) -> U13(X) 12.86/4.08 a__U21(X1, X2) -> U21(X1, X2) 12.86/4.08 a__U22(X) -> U22(X) 12.86/4.08 a__U31(X1, X2, X3) -> U31(X1, X2, X3) 12.86/4.08 a__U32(X1, X2) -> U32(X1, X2) 12.86/4.08 a__U33(X) -> U33(X) 12.86/4.08 a__U41(X1, X2) -> U41(X1, X2) 12.86/4.08 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 12.86/4.08 a__plus(X1, X2) -> plus(X1, X2) 12.86/4.08 a__U61(X) -> U61(X) 12.86/4.08 a__U71(X1, X2, X3) -> U71(X1, X2, X3) 12.86/4.08 a__x(X1, X2) -> x(X1, X2) 12.86/4.08 a__and(X1, X2) -> and(X1, X2) 12.86/4.08 a__isNatKind(X) -> isNatKind(X) 12.86/4.08 12.86/4.08 S is empty. 12.86/4.08 Rewrite Strategy: FULL 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (9) DecreasingLoopProof (FINISHED) 12.86/4.08 The following loop(s) give(s) rise to the lower bound EXP: 12.86/4.08 12.86/4.08 The rewrite sequence 12.86/4.08 12.86/4.08 mark(plus(X1, s(X1_0))) ->^+ a__U51(a__and(a__and(a__isNat(mark(X1_0)), isNatKind(mark(X1_0))), and(isNat(mark(X1)), isNatKind(mark(X1)))), mark(X1_0), mark(X1)) 12.86/4.08 12.86/4.08 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. 12.86/4.08 12.86/4.08 The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. 12.86/4.08 12.86/4.08 The result substitution is [ ]. 12.86/4.08 12.86/4.08 12.86/4.08 12.86/4.08 The rewrite sequence 12.86/4.08 12.86/4.08 mark(plus(X1, s(X1_0))) ->^+ a__U51(a__and(a__and(a__isNat(mark(X1_0)), isNatKind(mark(X1_0))), and(isNat(mark(X1)), isNatKind(mark(X1)))), mark(X1_0), mark(X1)) 12.86/4.08 12.86/4.08 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1,0]. 12.86/4.08 12.86/4.08 The pumping substitution is [X1_0 / plus(X1, s(X1_0))]. 12.86/4.08 12.86/4.08 The result substitution is [ ]. 12.86/4.08 12.86/4.08 12.86/4.08 12.86/4.08 12.86/4.08 ---------------------------------------- 12.86/4.08 12.86/4.08 (10) 12.86/4.08 BOUNDS(EXP, INF) 13.05/4.12 EOF