22.65/7.31 WORST_CASE(NON_POLY, ?) 22.65/7.32 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 22.65/7.32 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.65/7.32 22.65/7.32 22.65/7.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 22.65/7.32 22.65/7.32 (0) CpxTRS 22.65/7.32 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 22.65/7.32 (2) TRS for Loop Detection 22.65/7.32 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 22.65/7.32 (4) BEST 22.65/7.32 (5) proven lower bound 22.65/7.32 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 22.65/7.32 (7) BOUNDS(n^1, INF) 22.65/7.32 (8) TRS for Loop Detection 22.65/7.32 (9) DecreasingLoopProof [FINISHED, 4439 ms] 22.65/7.32 (10) BOUNDS(EXP, INF) 22.65/7.32 22.65/7.32 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (0) 22.65/7.32 Obligation: 22.65/7.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 22.65/7.32 22.65/7.32 22.65/7.32 The TRS R consists of the following rules: 22.65/7.32 22.65/7.32 a__zeros -> cons(0, zeros) 22.65/7.32 a__U11(tt, L) -> s(a__length(mark(L))) 22.65/7.32 a__U21(tt) -> nil 22.65/7.32 a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 22.65/7.32 a__and(tt, X) -> mark(X) 22.65/7.32 a__isNat(0) -> tt 22.65/7.32 a__isNat(length(V1)) -> a__isNatList(V1) 22.65/7.32 a__isNat(s(V1)) -> a__isNat(V1) 22.65/7.32 a__isNatIList(V) -> a__isNatList(V) 22.65/7.32 a__isNatIList(zeros) -> tt 22.65/7.32 a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__isNatList(nil) -> tt 22.65/7.32 a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) 22.65/7.32 a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__length(nil) -> 0 22.65/7.32 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) 22.65/7.32 a__take(0, IL) -> a__U21(a__isNatIList(IL)) 22.65/7.32 a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) 22.65/7.32 mark(zeros) -> a__zeros 22.65/7.32 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 22.65/7.32 mark(length(X)) -> a__length(mark(X)) 22.65/7.32 mark(U21(X)) -> a__U21(mark(X)) 22.65/7.32 mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) 22.65/7.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 22.65/7.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 22.65/7.32 mark(isNat(X)) -> a__isNat(X) 22.65/7.32 mark(isNatList(X)) -> a__isNatList(X) 22.65/7.32 mark(isNatIList(X)) -> a__isNatIList(X) 22.65/7.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 22.65/7.32 mark(0) -> 0 22.65/7.32 mark(tt) -> tt 22.65/7.32 mark(s(X)) -> s(mark(X)) 22.65/7.32 mark(nil) -> nil 22.65/7.32 a__zeros -> zeros 22.65/7.32 a__U11(X1, X2) -> U11(X1, X2) 22.65/7.32 a__length(X) -> length(X) 22.65/7.32 a__U21(X) -> U21(X) 22.65/7.32 a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) 22.65/7.32 a__take(X1, X2) -> take(X1, X2) 22.65/7.32 a__and(X1, X2) -> and(X1, X2) 22.65/7.32 a__isNat(X) -> isNat(X) 22.65/7.32 a__isNatList(X) -> isNatList(X) 22.65/7.32 a__isNatIList(X) -> isNatIList(X) 22.65/7.32 22.65/7.32 S is empty. 22.65/7.32 Rewrite Strategy: FULL 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 22.65/7.32 Transformed a relative TRS into a decreasing-loop problem. 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (2) 22.65/7.32 Obligation: 22.65/7.32 Analyzing the following TRS for decreasing loops: 22.65/7.32 22.65/7.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 22.65/7.32 22.65/7.32 22.65/7.32 The TRS R consists of the following rules: 22.65/7.32 22.65/7.32 a__zeros -> cons(0, zeros) 22.65/7.32 a__U11(tt, L) -> s(a__length(mark(L))) 22.65/7.32 a__U21(tt) -> nil 22.65/7.32 a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 22.65/7.32 a__and(tt, X) -> mark(X) 22.65/7.32 a__isNat(0) -> tt 22.65/7.32 a__isNat(length(V1)) -> a__isNatList(V1) 22.65/7.32 a__isNat(s(V1)) -> a__isNat(V1) 22.65/7.32 a__isNatIList(V) -> a__isNatList(V) 22.65/7.32 a__isNatIList(zeros) -> tt 22.65/7.32 a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__isNatList(nil) -> tt 22.65/7.32 a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) 22.65/7.32 a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__length(nil) -> 0 22.65/7.32 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) 22.65/7.32 a__take(0, IL) -> a__U21(a__isNatIList(IL)) 22.65/7.32 a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) 22.65/7.32 mark(zeros) -> a__zeros 22.65/7.32 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 22.65/7.32 mark(length(X)) -> a__length(mark(X)) 22.65/7.32 mark(U21(X)) -> a__U21(mark(X)) 22.65/7.32 mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) 22.65/7.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 22.65/7.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 22.65/7.32 mark(isNat(X)) -> a__isNat(X) 22.65/7.32 mark(isNatList(X)) -> a__isNatList(X) 22.65/7.32 mark(isNatIList(X)) -> a__isNatIList(X) 22.65/7.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 22.65/7.32 mark(0) -> 0 22.65/7.32 mark(tt) -> tt 22.65/7.32 mark(s(X)) -> s(mark(X)) 22.65/7.32 mark(nil) -> nil 22.65/7.32 a__zeros -> zeros 22.65/7.32 a__U11(X1, X2) -> U11(X1, X2) 22.65/7.32 a__length(X) -> length(X) 22.65/7.32 a__U21(X) -> U21(X) 22.65/7.32 a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) 22.65/7.32 a__take(X1, X2) -> take(X1, X2) 22.65/7.32 a__and(X1, X2) -> and(X1, X2) 22.65/7.32 a__isNat(X) -> isNat(X) 22.65/7.32 a__isNatList(X) -> isNatList(X) 22.65/7.32 a__isNatIList(X) -> isNatIList(X) 22.65/7.32 22.65/7.32 S is empty. 22.65/7.32 Rewrite Strategy: FULL 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (3) DecreasingLoopProof (LOWER BOUND(ID)) 22.65/7.32 The following loop(s) give(s) rise to the lower bound Omega(n^1): 22.65/7.32 22.65/7.32 The rewrite sequence 22.65/7.32 22.65/7.32 mark(length(X)) ->^+ a__length(mark(X)) 22.65/7.32 22.65/7.32 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 22.65/7.32 22.65/7.32 The pumping substitution is [X / length(X)]. 22.65/7.32 22.65/7.32 The result substitution is [ ]. 22.65/7.32 22.65/7.32 22.65/7.32 22.65/7.32 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (4) 22.65/7.32 Complex Obligation (BEST) 22.65/7.32 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (5) 22.65/7.32 Obligation: 22.65/7.32 Proved the lower bound n^1 for the following obligation: 22.65/7.32 22.65/7.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 22.65/7.32 22.65/7.32 22.65/7.32 The TRS R consists of the following rules: 22.65/7.32 22.65/7.32 a__zeros -> cons(0, zeros) 22.65/7.32 a__U11(tt, L) -> s(a__length(mark(L))) 22.65/7.32 a__U21(tt) -> nil 22.65/7.32 a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 22.65/7.32 a__and(tt, X) -> mark(X) 22.65/7.32 a__isNat(0) -> tt 22.65/7.32 a__isNat(length(V1)) -> a__isNatList(V1) 22.65/7.32 a__isNat(s(V1)) -> a__isNat(V1) 22.65/7.32 a__isNatIList(V) -> a__isNatList(V) 22.65/7.32 a__isNatIList(zeros) -> tt 22.65/7.32 a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__isNatList(nil) -> tt 22.65/7.32 a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) 22.65/7.32 a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__length(nil) -> 0 22.65/7.32 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) 22.65/7.32 a__take(0, IL) -> a__U21(a__isNatIList(IL)) 22.65/7.32 a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) 22.65/7.32 mark(zeros) -> a__zeros 22.65/7.32 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 22.65/7.32 mark(length(X)) -> a__length(mark(X)) 22.65/7.32 mark(U21(X)) -> a__U21(mark(X)) 22.65/7.32 mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) 22.65/7.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 22.65/7.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 22.65/7.32 mark(isNat(X)) -> a__isNat(X) 22.65/7.32 mark(isNatList(X)) -> a__isNatList(X) 22.65/7.32 mark(isNatIList(X)) -> a__isNatIList(X) 22.65/7.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 22.65/7.32 mark(0) -> 0 22.65/7.32 mark(tt) -> tt 22.65/7.32 mark(s(X)) -> s(mark(X)) 22.65/7.32 mark(nil) -> nil 22.65/7.32 a__zeros -> zeros 22.65/7.32 a__U11(X1, X2) -> U11(X1, X2) 22.65/7.32 a__length(X) -> length(X) 22.65/7.32 a__U21(X) -> U21(X) 22.65/7.32 a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) 22.65/7.32 a__take(X1, X2) -> take(X1, X2) 22.65/7.32 a__and(X1, X2) -> and(X1, X2) 22.65/7.32 a__isNat(X) -> isNat(X) 22.65/7.32 a__isNatList(X) -> isNatList(X) 22.65/7.32 a__isNatIList(X) -> isNatIList(X) 22.65/7.32 22.65/7.32 S is empty. 22.65/7.32 Rewrite Strategy: FULL 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (6) LowerBoundPropagationProof (FINISHED) 22.65/7.32 Propagated lower bound. 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (7) 22.65/7.32 BOUNDS(n^1, INF) 22.65/7.32 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (8) 22.65/7.32 Obligation: 22.65/7.32 Analyzing the following TRS for decreasing loops: 22.65/7.32 22.65/7.32 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 22.65/7.32 22.65/7.32 22.65/7.32 The TRS R consists of the following rules: 22.65/7.32 22.65/7.32 a__zeros -> cons(0, zeros) 22.65/7.32 a__U11(tt, L) -> s(a__length(mark(L))) 22.65/7.32 a__U21(tt) -> nil 22.65/7.32 a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 22.65/7.32 a__and(tt, X) -> mark(X) 22.65/7.32 a__isNat(0) -> tt 22.65/7.32 a__isNat(length(V1)) -> a__isNatList(V1) 22.65/7.32 a__isNat(s(V1)) -> a__isNat(V1) 22.65/7.32 a__isNatIList(V) -> a__isNatList(V) 22.65/7.32 a__isNatIList(zeros) -> tt 22.65/7.32 a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__isNatList(nil) -> tt 22.65/7.32 a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) 22.65/7.32 a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) 22.65/7.32 a__length(nil) -> 0 22.65/7.32 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) 22.65/7.32 a__take(0, IL) -> a__U21(a__isNatIList(IL)) 22.65/7.32 a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) 22.65/7.32 mark(zeros) -> a__zeros 22.65/7.32 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 22.65/7.32 mark(length(X)) -> a__length(mark(X)) 22.65/7.32 mark(U21(X)) -> a__U21(mark(X)) 22.65/7.32 mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) 22.65/7.32 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 22.65/7.32 mark(and(X1, X2)) -> a__and(mark(X1), X2) 22.65/7.32 mark(isNat(X)) -> a__isNat(X) 22.65/7.32 mark(isNatList(X)) -> a__isNatList(X) 22.65/7.32 mark(isNatIList(X)) -> a__isNatIList(X) 22.65/7.32 mark(cons(X1, X2)) -> cons(mark(X1), X2) 22.65/7.32 mark(0) -> 0 22.65/7.32 mark(tt) -> tt 22.65/7.32 mark(s(X)) -> s(mark(X)) 22.65/7.32 mark(nil) -> nil 22.65/7.32 a__zeros -> zeros 22.65/7.32 a__U11(X1, X2) -> U11(X1, X2) 22.65/7.32 a__length(X) -> length(X) 22.65/7.32 a__U21(X) -> U21(X) 22.65/7.32 a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) 22.65/7.32 a__take(X1, X2) -> take(X1, X2) 22.65/7.32 a__and(X1, X2) -> and(X1, X2) 22.65/7.32 a__isNat(X) -> isNat(X) 22.65/7.32 a__isNatList(X) -> isNatList(X) 22.65/7.32 a__isNatIList(X) -> isNatIList(X) 22.65/7.32 22.65/7.32 S is empty. 22.65/7.32 Rewrite Strategy: FULL 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (9) DecreasingLoopProof (FINISHED) 22.65/7.32 The following loop(s) give(s) rise to the lower bound EXP: 22.65/7.32 22.65/7.32 The rewrite sequence 22.65/7.32 22.65/7.32 mark(take(s(X1_0), cons(X11_1, X22_1))) ->^+ a__U31(a__and(a__isNatIList(X22_1), and(isNat(mark(X1_0)), isNat(mark(X11_1)))), X22_1, mark(X1_0), mark(X11_1)) 22.65/7.32 22.65/7.32 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0]. 22.65/7.32 22.65/7.32 The pumping substitution is [X1_0 / take(s(X1_0), cons(X11_1, X22_1))]. 22.65/7.32 22.65/7.32 The result substitution is [ ]. 22.65/7.32 22.65/7.32 22.65/7.32 22.65/7.32 The rewrite sequence 22.65/7.32 22.65/7.32 mark(take(s(X1_0), cons(X11_1, X22_1))) ->^+ a__U31(a__and(a__isNatIList(X22_1), and(isNat(mark(X1_0)), isNat(mark(X11_1)))), X22_1, mark(X1_0), mark(X11_1)) 22.65/7.32 22.65/7.32 gives rise to a decreasing loop by considering the right hand sides subterm at position [2]. 22.65/7.32 22.65/7.32 The pumping substitution is [X1_0 / take(s(X1_0), cons(X11_1, X22_1))]. 22.65/7.32 22.65/7.32 The result substitution is [ ]. 22.65/7.32 22.65/7.32 22.65/7.32 22.65/7.32 22.65/7.32 ---------------------------------------- 22.65/7.32 22.65/7.32 (10) 22.65/7.32 BOUNDS(EXP, INF) 22.88/7.39 EOF