1145.56/291.54 WORST_CASE(Omega(n^1), ?) 1158.33/294.71 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1158.33/294.71 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1158.33/294.71 1158.33/294.71 1158.33/294.71 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.33/294.71 1158.33/294.71 (0) CpxTRS 1158.33/294.71 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1158.33/294.71 (2) TRS for Loop Detection 1158.33/294.71 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1158.33/294.71 (4) BEST 1158.33/294.71 (5) proven lower bound 1158.33/294.71 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1158.33/294.71 (7) BOUNDS(n^1, INF) 1158.33/294.71 (8) TRS for Loop Detection 1158.33/294.71 1158.33/294.71 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (0) 1158.33/294.71 Obligation: 1158.33/294.71 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.33/294.71 1158.33/294.71 1158.33/294.71 The TRS R consists of the following rules: 1158.33/294.71 1158.33/294.71 active(zeros) -> mark(cons(0, zeros)) 1158.33/294.71 active(U11(tt, L)) -> mark(s(length(L))) 1158.33/294.71 active(U21(tt)) -> mark(nil) 1158.33/294.71 active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1158.33/294.71 active(and(tt, X)) -> mark(X) 1158.33/294.71 active(isNat(0)) -> mark(tt) 1158.33/294.71 active(isNat(length(V1))) -> mark(isNatList(V1)) 1158.33/294.71 active(isNat(s(V1))) -> mark(isNat(V1)) 1158.33/294.71 active(isNatIList(V)) -> mark(isNatList(V)) 1158.33/294.71 active(isNatIList(zeros)) -> mark(tt) 1158.33/294.71 active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(isNatList(nil)) -> mark(tt) 1158.33/294.71 active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) 1158.33/294.71 active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(length(nil)) -> mark(0) 1158.33/294.71 active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) 1158.33/294.71 active(take(0, IL)) -> mark(U21(isNatIList(IL))) 1158.33/294.71 active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) 1158.33/294.71 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.33/294.71 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.33/294.71 active(s(X)) -> s(active(X)) 1158.33/294.71 active(length(X)) -> length(active(X)) 1158.33/294.71 active(U21(X)) -> U21(active(X)) 1158.33/294.71 active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) 1158.33/294.71 active(take(X1, X2)) -> take(active(X1), X2) 1158.33/294.71 active(take(X1, X2)) -> take(X1, active(X2)) 1158.33/294.71 active(and(X1, X2)) -> and(active(X1), X2) 1158.33/294.71 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.33/294.71 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.33/294.71 s(mark(X)) -> mark(s(X)) 1158.33/294.71 length(mark(X)) -> mark(length(X)) 1158.33/294.71 U21(mark(X)) -> mark(U21(X)) 1158.33/294.71 U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) 1158.33/294.71 take(mark(X1), X2) -> mark(take(X1, X2)) 1158.33/294.71 take(X1, mark(X2)) -> mark(take(X1, X2)) 1158.33/294.71 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.33/294.71 proper(zeros) -> ok(zeros) 1158.33/294.71 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.33/294.71 proper(0) -> ok(0) 1158.33/294.71 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.33/294.71 proper(tt) -> ok(tt) 1158.33/294.71 proper(s(X)) -> s(proper(X)) 1158.33/294.71 proper(length(X)) -> length(proper(X)) 1158.33/294.71 proper(U21(X)) -> U21(proper(X)) 1158.33/294.71 proper(nil) -> ok(nil) 1158.33/294.71 proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) 1158.33/294.71 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1158.33/294.71 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.33/294.71 proper(isNat(X)) -> isNat(proper(X)) 1158.33/294.71 proper(isNatList(X)) -> isNatList(proper(X)) 1158.33/294.71 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.33/294.71 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.33/294.71 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.33/294.71 s(ok(X)) -> ok(s(X)) 1158.33/294.71 length(ok(X)) -> ok(length(X)) 1158.33/294.71 U21(ok(X)) -> ok(U21(X)) 1158.33/294.71 U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) 1158.33/294.71 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1158.33/294.71 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.33/294.71 isNat(ok(X)) -> ok(isNat(X)) 1158.33/294.71 isNatList(ok(X)) -> ok(isNatList(X)) 1158.33/294.71 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.33/294.71 top(mark(X)) -> top(proper(X)) 1158.33/294.71 top(ok(X)) -> top(active(X)) 1158.33/294.71 1158.33/294.71 S is empty. 1158.33/294.71 Rewrite Strategy: FULL 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1158.33/294.71 Transformed a relative TRS into a decreasing-loop problem. 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (2) 1158.33/294.71 Obligation: 1158.33/294.71 Analyzing the following TRS for decreasing loops: 1158.33/294.71 1158.33/294.71 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.33/294.71 1158.33/294.71 1158.33/294.71 The TRS R consists of the following rules: 1158.33/294.71 1158.33/294.71 active(zeros) -> mark(cons(0, zeros)) 1158.33/294.71 active(U11(tt, L)) -> mark(s(length(L))) 1158.33/294.71 active(U21(tt)) -> mark(nil) 1158.33/294.71 active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1158.33/294.71 active(and(tt, X)) -> mark(X) 1158.33/294.71 active(isNat(0)) -> mark(tt) 1158.33/294.71 active(isNat(length(V1))) -> mark(isNatList(V1)) 1158.33/294.71 active(isNat(s(V1))) -> mark(isNat(V1)) 1158.33/294.71 active(isNatIList(V)) -> mark(isNatList(V)) 1158.33/294.71 active(isNatIList(zeros)) -> mark(tt) 1158.33/294.71 active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(isNatList(nil)) -> mark(tt) 1158.33/294.71 active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) 1158.33/294.71 active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(length(nil)) -> mark(0) 1158.33/294.71 active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) 1158.33/294.71 active(take(0, IL)) -> mark(U21(isNatIList(IL))) 1158.33/294.71 active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) 1158.33/294.71 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.33/294.71 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.33/294.71 active(s(X)) -> s(active(X)) 1158.33/294.71 active(length(X)) -> length(active(X)) 1158.33/294.71 active(U21(X)) -> U21(active(X)) 1158.33/294.71 active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) 1158.33/294.71 active(take(X1, X2)) -> take(active(X1), X2) 1158.33/294.71 active(take(X1, X2)) -> take(X1, active(X2)) 1158.33/294.71 active(and(X1, X2)) -> and(active(X1), X2) 1158.33/294.71 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.33/294.71 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.33/294.71 s(mark(X)) -> mark(s(X)) 1158.33/294.71 length(mark(X)) -> mark(length(X)) 1158.33/294.71 U21(mark(X)) -> mark(U21(X)) 1158.33/294.71 U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) 1158.33/294.71 take(mark(X1), X2) -> mark(take(X1, X2)) 1158.33/294.71 take(X1, mark(X2)) -> mark(take(X1, X2)) 1158.33/294.71 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.33/294.71 proper(zeros) -> ok(zeros) 1158.33/294.71 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.33/294.71 proper(0) -> ok(0) 1158.33/294.71 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.33/294.71 proper(tt) -> ok(tt) 1158.33/294.71 proper(s(X)) -> s(proper(X)) 1158.33/294.71 proper(length(X)) -> length(proper(X)) 1158.33/294.71 proper(U21(X)) -> U21(proper(X)) 1158.33/294.71 proper(nil) -> ok(nil) 1158.33/294.71 proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) 1158.33/294.71 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1158.33/294.71 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.33/294.71 proper(isNat(X)) -> isNat(proper(X)) 1158.33/294.71 proper(isNatList(X)) -> isNatList(proper(X)) 1158.33/294.71 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.33/294.71 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.33/294.71 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.33/294.71 s(ok(X)) -> ok(s(X)) 1158.33/294.71 length(ok(X)) -> ok(length(X)) 1158.33/294.71 U21(ok(X)) -> ok(U21(X)) 1158.33/294.71 U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) 1158.33/294.71 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1158.33/294.71 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.33/294.71 isNat(ok(X)) -> ok(isNat(X)) 1158.33/294.71 isNatList(ok(X)) -> ok(isNatList(X)) 1158.33/294.71 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.33/294.71 top(mark(X)) -> top(proper(X)) 1158.33/294.71 top(ok(X)) -> top(active(X)) 1158.33/294.71 1158.33/294.71 S is empty. 1158.33/294.71 Rewrite Strategy: FULL 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1158.33/294.71 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1158.33/294.71 1158.33/294.71 The rewrite sequence 1158.33/294.71 1158.33/294.71 take(ok(X1), ok(X2)) ->^+ ok(take(X1, X2)) 1158.33/294.71 1158.33/294.71 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1158.33/294.71 1158.33/294.71 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 1158.33/294.71 1158.33/294.71 The result substitution is [ ]. 1158.33/294.71 1158.33/294.71 1158.33/294.71 1158.33/294.71 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (4) 1158.33/294.71 Complex Obligation (BEST) 1158.33/294.71 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (5) 1158.33/294.71 Obligation: 1158.33/294.71 Proved the lower bound n^1 for the following obligation: 1158.33/294.71 1158.33/294.71 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.33/294.71 1158.33/294.71 1158.33/294.71 The TRS R consists of the following rules: 1158.33/294.71 1158.33/294.71 active(zeros) -> mark(cons(0, zeros)) 1158.33/294.71 active(U11(tt, L)) -> mark(s(length(L))) 1158.33/294.71 active(U21(tt)) -> mark(nil) 1158.33/294.71 active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1158.33/294.71 active(and(tt, X)) -> mark(X) 1158.33/294.71 active(isNat(0)) -> mark(tt) 1158.33/294.71 active(isNat(length(V1))) -> mark(isNatList(V1)) 1158.33/294.71 active(isNat(s(V1))) -> mark(isNat(V1)) 1158.33/294.71 active(isNatIList(V)) -> mark(isNatList(V)) 1158.33/294.71 active(isNatIList(zeros)) -> mark(tt) 1158.33/294.71 active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(isNatList(nil)) -> mark(tt) 1158.33/294.71 active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) 1158.33/294.71 active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(length(nil)) -> mark(0) 1158.33/294.71 active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) 1158.33/294.71 active(take(0, IL)) -> mark(U21(isNatIList(IL))) 1158.33/294.71 active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) 1158.33/294.71 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.33/294.71 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.33/294.71 active(s(X)) -> s(active(X)) 1158.33/294.71 active(length(X)) -> length(active(X)) 1158.33/294.71 active(U21(X)) -> U21(active(X)) 1158.33/294.71 active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) 1158.33/294.71 active(take(X1, X2)) -> take(active(X1), X2) 1158.33/294.71 active(take(X1, X2)) -> take(X1, active(X2)) 1158.33/294.71 active(and(X1, X2)) -> and(active(X1), X2) 1158.33/294.71 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.33/294.71 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.33/294.71 s(mark(X)) -> mark(s(X)) 1158.33/294.71 length(mark(X)) -> mark(length(X)) 1158.33/294.71 U21(mark(X)) -> mark(U21(X)) 1158.33/294.71 U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) 1158.33/294.71 take(mark(X1), X2) -> mark(take(X1, X2)) 1158.33/294.71 take(X1, mark(X2)) -> mark(take(X1, X2)) 1158.33/294.71 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.33/294.71 proper(zeros) -> ok(zeros) 1158.33/294.71 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.33/294.71 proper(0) -> ok(0) 1158.33/294.71 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.33/294.71 proper(tt) -> ok(tt) 1158.33/294.71 proper(s(X)) -> s(proper(X)) 1158.33/294.71 proper(length(X)) -> length(proper(X)) 1158.33/294.71 proper(U21(X)) -> U21(proper(X)) 1158.33/294.71 proper(nil) -> ok(nil) 1158.33/294.71 proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) 1158.33/294.71 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1158.33/294.71 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.33/294.71 proper(isNat(X)) -> isNat(proper(X)) 1158.33/294.71 proper(isNatList(X)) -> isNatList(proper(X)) 1158.33/294.71 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.33/294.71 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.33/294.71 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.33/294.71 s(ok(X)) -> ok(s(X)) 1158.33/294.71 length(ok(X)) -> ok(length(X)) 1158.33/294.71 U21(ok(X)) -> ok(U21(X)) 1158.33/294.71 U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) 1158.33/294.71 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1158.33/294.71 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.33/294.71 isNat(ok(X)) -> ok(isNat(X)) 1158.33/294.71 isNatList(ok(X)) -> ok(isNatList(X)) 1158.33/294.71 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.33/294.71 top(mark(X)) -> top(proper(X)) 1158.33/294.71 top(ok(X)) -> top(active(X)) 1158.33/294.71 1158.33/294.71 S is empty. 1158.33/294.71 Rewrite Strategy: FULL 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (6) LowerBoundPropagationProof (FINISHED) 1158.33/294.71 Propagated lower bound. 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (7) 1158.33/294.71 BOUNDS(n^1, INF) 1158.33/294.71 1158.33/294.71 ---------------------------------------- 1158.33/294.71 1158.33/294.71 (8) 1158.33/294.71 Obligation: 1158.33/294.71 Analyzing the following TRS for decreasing loops: 1158.33/294.71 1158.33/294.71 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1158.33/294.71 1158.33/294.71 1158.33/294.71 The TRS R consists of the following rules: 1158.33/294.71 1158.33/294.71 active(zeros) -> mark(cons(0, zeros)) 1158.33/294.71 active(U11(tt, L)) -> mark(s(length(L))) 1158.33/294.71 active(U21(tt)) -> mark(nil) 1158.33/294.71 active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1158.33/294.71 active(and(tt, X)) -> mark(X) 1158.33/294.71 active(isNat(0)) -> mark(tt) 1158.33/294.71 active(isNat(length(V1))) -> mark(isNatList(V1)) 1158.33/294.71 active(isNat(s(V1))) -> mark(isNat(V1)) 1158.33/294.71 active(isNatIList(V)) -> mark(isNatList(V)) 1158.33/294.71 active(isNatIList(zeros)) -> mark(tt) 1158.33/294.71 active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(isNatList(nil)) -> mark(tt) 1158.33/294.71 active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) 1158.33/294.71 active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) 1158.33/294.71 active(length(nil)) -> mark(0) 1158.33/294.71 active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) 1158.33/294.71 active(take(0, IL)) -> mark(U21(isNatIList(IL))) 1158.33/294.71 active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) 1158.33/294.71 active(cons(X1, X2)) -> cons(active(X1), X2) 1158.33/294.71 active(U11(X1, X2)) -> U11(active(X1), X2) 1158.33/294.71 active(s(X)) -> s(active(X)) 1158.33/294.71 active(length(X)) -> length(active(X)) 1158.33/294.71 active(U21(X)) -> U21(active(X)) 1158.33/294.71 active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) 1158.33/294.72 active(take(X1, X2)) -> take(active(X1), X2) 1158.33/294.72 active(take(X1, X2)) -> take(X1, active(X2)) 1158.33/294.72 active(and(X1, X2)) -> and(active(X1), X2) 1158.33/294.72 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1158.33/294.72 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1158.33/294.72 s(mark(X)) -> mark(s(X)) 1158.33/294.72 length(mark(X)) -> mark(length(X)) 1158.33/294.72 U21(mark(X)) -> mark(U21(X)) 1158.33/294.72 U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) 1158.33/294.72 take(mark(X1), X2) -> mark(take(X1, X2)) 1158.33/294.72 take(X1, mark(X2)) -> mark(take(X1, X2)) 1158.33/294.72 and(mark(X1), X2) -> mark(and(X1, X2)) 1158.33/294.72 proper(zeros) -> ok(zeros) 1158.33/294.72 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1158.33/294.72 proper(0) -> ok(0) 1158.33/294.72 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1158.33/294.72 proper(tt) -> ok(tt) 1158.33/294.72 proper(s(X)) -> s(proper(X)) 1158.33/294.72 proper(length(X)) -> length(proper(X)) 1158.33/294.72 proper(U21(X)) -> U21(proper(X)) 1158.33/294.72 proper(nil) -> ok(nil) 1158.33/294.72 proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) 1158.33/294.72 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1158.33/294.72 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1158.33/294.72 proper(isNat(X)) -> isNat(proper(X)) 1158.33/294.72 proper(isNatList(X)) -> isNatList(proper(X)) 1158.33/294.72 proper(isNatIList(X)) -> isNatIList(proper(X)) 1158.33/294.72 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1158.33/294.72 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1158.33/294.72 s(ok(X)) -> ok(s(X)) 1158.33/294.72 length(ok(X)) -> ok(length(X)) 1158.33/294.72 U21(ok(X)) -> ok(U21(X)) 1158.33/294.72 U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) 1158.33/294.72 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1158.33/294.72 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1158.33/294.72 isNat(ok(X)) -> ok(isNat(X)) 1158.33/294.72 isNatList(ok(X)) -> ok(isNatList(X)) 1158.33/294.72 isNatIList(ok(X)) -> ok(isNatIList(X)) 1158.33/294.72 top(mark(X)) -> top(proper(X)) 1158.33/294.72 top(ok(X)) -> top(active(X)) 1158.33/294.72 1158.33/294.72 S is empty. 1158.33/294.72 Rewrite Strategy: FULL 1158.43/294.78 EOF