3.55/1.73 WORST_CASE(NON_POLY, ?) 3.85/1.74 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.85/1.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.85/1.74 3.85/1.74 3.85/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.85/1.74 3.85/1.74 (0) CpxTRS 3.85/1.74 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.85/1.74 (2) TRS for Loop Detection 3.85/1.74 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.85/1.74 (4) BEST 3.85/1.74 (5) proven lower bound 3.85/1.74 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 3.85/1.74 (7) BOUNDS(n^1, INF) 3.85/1.74 (8) TRS for Loop Detection 3.85/1.74 (9) DecreasingLoopProof [FINISHED, 13 ms] 3.85/1.74 (10) BOUNDS(EXP, INF) 3.85/1.74 3.85/1.74 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (0) 3.85/1.74 Obligation: 3.85/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.85/1.74 3.85/1.74 3.85/1.74 The TRS R consists of the following rules: 3.85/1.74 3.85/1.74 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 3.85/1.74 a__sqr(0) -> 0 3.85/1.74 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 3.85/1.74 a__dbl(0) -> 0 3.85/1.74 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 3.85/1.74 a__add(0, X) -> mark(X) 3.85/1.74 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 3.85/1.74 a__first(0, X) -> nil 3.85/1.74 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 3.85/1.74 a__half(0) -> 0 3.85/1.74 a__half(s(0)) -> 0 3.85/1.74 a__half(s(s(X))) -> s(a__half(mark(X))) 3.85/1.74 a__half(dbl(X)) -> mark(X) 3.85/1.74 mark(terms(X)) -> a__terms(mark(X)) 3.85/1.74 mark(sqr(X)) -> a__sqr(mark(X)) 3.85/1.74 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 3.85/1.74 mark(dbl(X)) -> a__dbl(mark(X)) 3.85/1.74 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 3.85/1.74 mark(half(X)) -> a__half(mark(X)) 3.85/1.74 mark(cons(X1, X2)) -> cons(mark(X1), X2) 3.85/1.74 mark(recip(X)) -> recip(mark(X)) 3.85/1.74 mark(s(X)) -> s(mark(X)) 3.85/1.74 mark(0) -> 0 3.85/1.74 mark(nil) -> nil 3.85/1.74 a__terms(X) -> terms(X) 3.85/1.74 a__sqr(X) -> sqr(X) 3.85/1.74 a__add(X1, X2) -> add(X1, X2) 3.85/1.74 a__dbl(X) -> dbl(X) 3.85/1.74 a__first(X1, X2) -> first(X1, X2) 3.85/1.74 a__half(X) -> half(X) 3.85/1.74 3.85/1.74 S is empty. 3.85/1.74 Rewrite Strategy: FULL 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.85/1.74 Transformed a relative TRS into a decreasing-loop problem. 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (2) 3.85/1.74 Obligation: 3.85/1.74 Analyzing the following TRS for decreasing loops: 3.85/1.74 3.85/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.85/1.74 3.85/1.74 3.85/1.74 The TRS R consists of the following rules: 3.85/1.74 3.85/1.74 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 3.85/1.74 a__sqr(0) -> 0 3.85/1.74 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 3.85/1.74 a__dbl(0) -> 0 3.85/1.74 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 3.85/1.74 a__add(0, X) -> mark(X) 3.85/1.74 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 3.85/1.74 a__first(0, X) -> nil 3.85/1.74 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 3.85/1.74 a__half(0) -> 0 3.85/1.74 a__half(s(0)) -> 0 3.85/1.74 a__half(s(s(X))) -> s(a__half(mark(X))) 3.85/1.74 a__half(dbl(X)) -> mark(X) 3.85/1.74 mark(terms(X)) -> a__terms(mark(X)) 3.85/1.74 mark(sqr(X)) -> a__sqr(mark(X)) 3.85/1.74 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 3.85/1.74 mark(dbl(X)) -> a__dbl(mark(X)) 3.85/1.74 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 3.85/1.74 mark(half(X)) -> a__half(mark(X)) 3.85/1.74 mark(cons(X1, X2)) -> cons(mark(X1), X2) 3.85/1.74 mark(recip(X)) -> recip(mark(X)) 3.85/1.74 mark(s(X)) -> s(mark(X)) 3.85/1.74 mark(0) -> 0 3.85/1.74 mark(nil) -> nil 3.85/1.74 a__terms(X) -> terms(X) 3.85/1.74 a__sqr(X) -> sqr(X) 3.85/1.74 a__add(X1, X2) -> add(X1, X2) 3.85/1.74 a__dbl(X) -> dbl(X) 3.85/1.74 a__first(X1, X2) -> first(X1, X2) 3.85/1.74 a__half(X) -> half(X) 3.85/1.74 3.85/1.74 S is empty. 3.85/1.74 Rewrite Strategy: FULL 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (3) DecreasingLoopProof (LOWER BOUND(ID)) 3.85/1.74 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.85/1.74 3.85/1.74 The rewrite sequence 3.85/1.74 3.85/1.74 mark(terms(X)) ->^+ a__terms(mark(X)) 3.85/1.74 3.85/1.74 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 3.85/1.74 3.85/1.74 The pumping substitution is [X / terms(X)]. 3.85/1.74 3.85/1.74 The result substitution is [ ]. 3.85/1.74 3.85/1.74 3.85/1.74 3.85/1.74 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (4) 3.85/1.74 Complex Obligation (BEST) 3.85/1.74 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (5) 3.85/1.74 Obligation: 3.85/1.74 Proved the lower bound n^1 for the following obligation: 3.85/1.74 3.85/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.85/1.74 3.85/1.74 3.85/1.74 The TRS R consists of the following rules: 3.85/1.74 3.85/1.74 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 3.85/1.74 a__sqr(0) -> 0 3.85/1.74 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 3.85/1.74 a__dbl(0) -> 0 3.85/1.74 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 3.85/1.74 a__add(0, X) -> mark(X) 3.85/1.74 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 3.85/1.74 a__first(0, X) -> nil 3.85/1.74 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 3.85/1.74 a__half(0) -> 0 3.85/1.74 a__half(s(0)) -> 0 3.85/1.74 a__half(s(s(X))) -> s(a__half(mark(X))) 3.85/1.74 a__half(dbl(X)) -> mark(X) 3.85/1.74 mark(terms(X)) -> a__terms(mark(X)) 3.85/1.74 mark(sqr(X)) -> a__sqr(mark(X)) 3.85/1.74 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 3.85/1.74 mark(dbl(X)) -> a__dbl(mark(X)) 3.85/1.74 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 3.85/1.74 mark(half(X)) -> a__half(mark(X)) 3.85/1.74 mark(cons(X1, X2)) -> cons(mark(X1), X2) 3.85/1.74 mark(recip(X)) -> recip(mark(X)) 3.85/1.74 mark(s(X)) -> s(mark(X)) 3.85/1.74 mark(0) -> 0 3.85/1.74 mark(nil) -> nil 3.85/1.74 a__terms(X) -> terms(X) 3.85/1.74 a__sqr(X) -> sqr(X) 3.85/1.74 a__add(X1, X2) -> add(X1, X2) 3.85/1.74 a__dbl(X) -> dbl(X) 3.85/1.74 a__first(X1, X2) -> first(X1, X2) 3.85/1.74 a__half(X) -> half(X) 3.85/1.74 3.85/1.74 S is empty. 3.85/1.74 Rewrite Strategy: FULL 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (6) LowerBoundPropagationProof (FINISHED) 3.85/1.74 Propagated lower bound. 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (7) 3.85/1.74 BOUNDS(n^1, INF) 3.85/1.74 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (8) 3.85/1.74 Obligation: 3.85/1.74 Analyzing the following TRS for decreasing loops: 3.85/1.74 3.85/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 3.85/1.74 3.85/1.74 3.85/1.74 The TRS R consists of the following rules: 3.85/1.74 3.85/1.74 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 3.85/1.74 a__sqr(0) -> 0 3.85/1.74 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 3.85/1.74 a__dbl(0) -> 0 3.85/1.74 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 3.85/1.74 a__add(0, X) -> mark(X) 3.85/1.74 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 3.85/1.74 a__first(0, X) -> nil 3.85/1.74 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 3.85/1.74 a__half(0) -> 0 3.85/1.74 a__half(s(0)) -> 0 3.85/1.74 a__half(s(s(X))) -> s(a__half(mark(X))) 3.85/1.74 a__half(dbl(X)) -> mark(X) 3.85/1.74 mark(terms(X)) -> a__terms(mark(X)) 3.85/1.74 mark(sqr(X)) -> a__sqr(mark(X)) 3.85/1.74 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 3.85/1.74 mark(dbl(X)) -> a__dbl(mark(X)) 3.85/1.74 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 3.85/1.74 mark(half(X)) -> a__half(mark(X)) 3.85/1.74 mark(cons(X1, X2)) -> cons(mark(X1), X2) 3.85/1.74 mark(recip(X)) -> recip(mark(X)) 3.85/1.74 mark(s(X)) -> s(mark(X)) 3.85/1.74 mark(0) -> 0 3.85/1.74 mark(nil) -> nil 3.85/1.74 a__terms(X) -> terms(X) 3.85/1.74 a__sqr(X) -> sqr(X) 3.85/1.74 a__add(X1, X2) -> add(X1, X2) 3.85/1.74 a__dbl(X) -> dbl(X) 3.85/1.74 a__first(X1, X2) -> first(X1, X2) 3.85/1.74 a__half(X) -> half(X) 3.85/1.74 3.85/1.74 S is empty. 3.85/1.74 Rewrite Strategy: FULL 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (9) DecreasingLoopProof (FINISHED) 3.85/1.74 The following loop(s) give(s) rise to the lower bound EXP: 3.85/1.74 3.85/1.74 The rewrite sequence 3.85/1.74 3.85/1.74 mark(terms(X)) ->^+ cons(recip(a__sqr(mark(mark(X)))), terms(s(mark(X)))) 3.85/1.74 3.85/1.74 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. 3.85/1.74 3.85/1.74 The pumping substitution is [X / terms(X)]. 3.85/1.74 3.85/1.74 The result substitution is [ ]. 3.85/1.74 3.85/1.74 3.85/1.74 3.85/1.74 The rewrite sequence 3.85/1.74 3.85/1.74 mark(terms(X)) ->^+ cons(recip(a__sqr(mark(mark(X)))), terms(s(mark(X)))) 3.85/1.74 3.85/1.74 gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. 3.85/1.74 3.85/1.74 The pumping substitution is [X / terms(X)]. 3.85/1.74 3.85/1.74 The result substitution is [ ]. 3.85/1.74 3.85/1.74 3.85/1.74 3.85/1.74 3.85/1.74 ---------------------------------------- 3.85/1.74 3.85/1.74 (10) 3.85/1.74 BOUNDS(EXP, INF) 3.88/1.78 EOF