2.81/1.58 WORST_CASE(NON_POLY, ?) 3.19/1.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.19/1.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.19/1.58 3.19/1.58 3.19/1.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 3.19/1.58 3.19/1.58 (0) CpxTRS 3.19/1.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.19/1.58 (2) TRS for Loop Detection 3.19/1.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.19/1.58 (4) BEST 3.19/1.58 (5) proven lower bound 3.19/1.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 3.19/1.58 (7) BOUNDS(n^1, INF) 3.19/1.58 (8) TRS for Loop Detection 3.19/1.58 (9) InfiniteLowerBoundProof [FINISHED, 0 ms] 3.19/1.58 (10) BOUNDS(INF, INF) 3.19/1.58 3.19/1.58 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (0) 3.19/1.58 Obligation: 3.19/1.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 3.19/1.58 3.19/1.58 3.19/1.58 The TRS R consists of the following rules: 3.19/1.58 3.19/1.58 terms(N) -> cons(recip(sqr(N)), terms(s(N))) 3.19/1.58 sqr(0) -> 0 3.19/1.58 sqr(s(X)) -> s(add(sqr(X), dbl(X))) 3.19/1.58 dbl(0) -> 0 3.19/1.58 dbl(s(X)) -> s(s(dbl(X))) 3.19/1.58 add(0, X) -> X 3.19/1.58 add(s(X), Y) -> s(add(X, Y)) 3.19/1.58 first(0, X) -> nil 3.19/1.58 first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 3.19/1.58 3.19/1.58 S is empty. 3.19/1.58 Rewrite Strategy: FULL 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.19/1.58 Transformed a relative TRS into a decreasing-loop problem. 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (2) 3.19/1.58 Obligation: 3.19/1.58 Analyzing the following TRS for decreasing loops: 3.19/1.58 3.19/1.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 3.19/1.58 3.19/1.58 3.19/1.58 The TRS R consists of the following rules: 3.19/1.58 3.19/1.58 terms(N) -> cons(recip(sqr(N)), terms(s(N))) 3.19/1.58 sqr(0) -> 0 3.19/1.58 sqr(s(X)) -> s(add(sqr(X), dbl(X))) 3.19/1.58 dbl(0) -> 0 3.19/1.58 dbl(s(X)) -> s(s(dbl(X))) 3.19/1.58 add(0, X) -> X 3.19/1.58 add(s(X), Y) -> s(add(X, Y)) 3.19/1.58 first(0, X) -> nil 3.19/1.58 first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 3.19/1.58 3.19/1.58 S is empty. 3.19/1.58 Rewrite Strategy: FULL 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 3.19/1.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.19/1.58 3.19/1.58 The rewrite sequence 3.19/1.58 3.19/1.58 first(s(X), cons(Y, Z)) ->^+ cons(Y, first(X, Z)) 3.19/1.58 3.19/1.58 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 3.19/1.58 3.19/1.58 The pumping substitution is [X / s(X), Z / cons(Y, Z)]. 3.19/1.58 3.19/1.58 The result substitution is [ ]. 3.19/1.58 3.19/1.58 3.19/1.58 3.19/1.58 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (4) 3.19/1.58 Complex Obligation (BEST) 3.19/1.58 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (5) 3.19/1.58 Obligation: 3.19/1.58 Proved the lower bound n^1 for the following obligation: 3.19/1.58 3.19/1.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 3.19/1.58 3.19/1.58 3.19/1.58 The TRS R consists of the following rules: 3.19/1.58 3.19/1.58 terms(N) -> cons(recip(sqr(N)), terms(s(N))) 3.19/1.58 sqr(0) -> 0 3.19/1.58 sqr(s(X)) -> s(add(sqr(X), dbl(X))) 3.19/1.58 dbl(0) -> 0 3.19/1.58 dbl(s(X)) -> s(s(dbl(X))) 3.19/1.58 add(0, X) -> X 3.19/1.58 add(s(X), Y) -> s(add(X, Y)) 3.19/1.58 first(0, X) -> nil 3.19/1.58 first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 3.19/1.58 3.19/1.58 S is empty. 3.19/1.58 Rewrite Strategy: FULL 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (6) LowerBoundPropagationProof (FINISHED) 3.19/1.58 Propagated lower bound. 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (7) 3.19/1.58 BOUNDS(n^1, INF) 3.19/1.58 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (8) 3.19/1.58 Obligation: 3.19/1.58 Analyzing the following TRS for decreasing loops: 3.19/1.58 3.19/1.58 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 3.19/1.58 3.19/1.58 3.19/1.58 The TRS R consists of the following rules: 3.19/1.58 3.19/1.58 terms(N) -> cons(recip(sqr(N)), terms(s(N))) 3.19/1.58 sqr(0) -> 0 3.19/1.58 sqr(s(X)) -> s(add(sqr(X), dbl(X))) 3.19/1.58 dbl(0) -> 0 3.19/1.58 dbl(s(X)) -> s(s(dbl(X))) 3.19/1.58 add(0, X) -> X 3.19/1.58 add(s(X), Y) -> s(add(X, Y)) 3.19/1.58 first(0, X) -> nil 3.19/1.58 first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 3.19/1.58 3.19/1.58 S is empty. 3.19/1.58 Rewrite Strategy: FULL 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (9) InfiniteLowerBoundProof (FINISHED) 3.19/1.58 The following loop proves infinite runtime complexity: 3.19/1.58 3.19/1.58 The rewrite sequence 3.19/1.58 3.19/1.58 terms(N) ->^+ cons(recip(sqr(N)), terms(s(N))) 3.19/1.58 3.19/1.58 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 3.19/1.58 3.19/1.58 The pumping substitution is [ ]. 3.19/1.58 3.19/1.58 The result substitution is [N / s(N)]. 3.19/1.58 3.19/1.58 3.19/1.58 3.19/1.58 3.19/1.58 ---------------------------------------- 3.19/1.58 3.19/1.58 (10) 3.19/1.58 BOUNDS(INF, INF) 3.22/1.63 EOF