3.54/1.72	WORST_CASE(NON_POLY, ?)
3.54/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.54/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.54/1.73	
3.54/1.73	
3.54/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.54/1.73	
3.54/1.73	(0) CpxTRS
3.54/1.73	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.54/1.73	(2) TRS for Loop Detection
3.54/1.73	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.54/1.73	(4) BEST
3.54/1.73	    (5) proven lower bound
3.54/1.73	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.54/1.73	        (7) BOUNDS(n^1, INF)
3.54/1.73	    (8) TRS for Loop Detection
3.54/1.73	        (9) DecreasingLoopProof [FINISHED, 26 ms]
3.54/1.73	        (10) BOUNDS(EXP, INF)
3.54/1.73	
3.54/1.73	
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(0)
3.54/1.73	Obligation:
3.54/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.54/1.73	
3.54/1.73	
3.54/1.73	The TRS R consists of the following rules:
3.54/1.73	
3.54/1.73	   fact(X) -> if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X))))
3.54/1.73	   add(0, X) -> X
3.54/1.73	   add(s(X), Y) -> s(add(X, Y))
3.54/1.73	   prod(0, X) -> 0
3.54/1.73	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.54/1.73	   if(true, X, Y) -> activate(X)
3.54/1.73	   if(false, X, Y) -> activate(Y)
3.54/1.73	   zero(0) -> true
3.54/1.73	   zero(s(X)) -> false
3.54/1.73	   p(s(X)) -> X
3.54/1.73	   s(X) -> n__s(X)
3.54/1.73	   0 -> n__0
3.54/1.73	   prod(X1, X2) -> n__prod(X1, X2)
3.54/1.73	   fact(X) -> n__fact(X)
3.54/1.73	   p(X) -> n__p(X)
3.54/1.73	   activate(n__s(X)) -> s(activate(X))
3.54/1.73	   activate(n__0) -> 0
3.54/1.73	   activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2))
3.54/1.73	   activate(n__fact(X)) -> fact(activate(X))
3.54/1.73	   activate(n__p(X)) -> p(activate(X))
3.54/1.73	   activate(X) -> X
3.54/1.73	
3.54/1.73	S is empty.
3.54/1.73	Rewrite Strategy: FULL
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.54/1.73	Transformed a relative TRS into a decreasing-loop problem.
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(2)
3.54/1.73	Obligation:
3.54/1.73	Analyzing the following TRS for decreasing loops:
3.54/1.73	
3.54/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.54/1.73	
3.54/1.73	
3.54/1.73	The TRS R consists of the following rules:
3.54/1.73	
3.54/1.73	   fact(X) -> if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X))))
3.54/1.73	   add(0, X) -> X
3.54/1.73	   add(s(X), Y) -> s(add(X, Y))
3.54/1.73	   prod(0, X) -> 0
3.54/1.73	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.54/1.73	   if(true, X, Y) -> activate(X)
3.54/1.73	   if(false, X, Y) -> activate(Y)
3.54/1.73	   zero(0) -> true
3.54/1.73	   zero(s(X)) -> false
3.54/1.73	   p(s(X)) -> X
3.54/1.73	   s(X) -> n__s(X)
3.54/1.73	   0 -> n__0
3.54/1.73	   prod(X1, X2) -> n__prod(X1, X2)
3.54/1.73	   fact(X) -> n__fact(X)
3.54/1.73	   p(X) -> n__p(X)
3.54/1.73	   activate(n__s(X)) -> s(activate(X))
3.54/1.73	   activate(n__0) -> 0
3.54/1.73	   activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2))
3.54/1.73	   activate(n__fact(X)) -> fact(activate(X))
3.54/1.73	   activate(n__p(X)) -> p(activate(X))
3.54/1.73	   activate(X) -> X
3.54/1.73	
3.54/1.73	S is empty.
3.54/1.73	Rewrite Strategy: FULL
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.54/1.73	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.54/1.73	
3.54/1.73	The rewrite sequence
3.54/1.73	
3.54/1.73	activate(n__p(X)) ->^+ p(activate(X))
3.54/1.73	
3.54/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.54/1.73	
3.54/1.73	The pumping substitution is [X / n__p(X)].
3.54/1.73	
3.54/1.73	The result substitution is [ ].
3.54/1.73	
3.54/1.73	
3.54/1.73	
3.54/1.73	
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(4)
3.54/1.73	Complex Obligation (BEST)
3.54/1.73	
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(5)
3.54/1.73	Obligation:
3.54/1.73	Proved the lower bound n^1 for the following obligation:
3.54/1.73	
3.54/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.54/1.73	
3.54/1.73	
3.54/1.73	The TRS R consists of the following rules:
3.54/1.73	
3.54/1.73	   fact(X) -> if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X))))
3.54/1.73	   add(0, X) -> X
3.54/1.73	   add(s(X), Y) -> s(add(X, Y))
3.54/1.73	   prod(0, X) -> 0
3.54/1.73	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.54/1.73	   if(true, X, Y) -> activate(X)
3.54/1.73	   if(false, X, Y) -> activate(Y)
3.54/1.73	   zero(0) -> true
3.54/1.73	   zero(s(X)) -> false
3.54/1.73	   p(s(X)) -> X
3.54/1.73	   s(X) -> n__s(X)
3.54/1.73	   0 -> n__0
3.54/1.73	   prod(X1, X2) -> n__prod(X1, X2)
3.54/1.73	   fact(X) -> n__fact(X)
3.54/1.73	   p(X) -> n__p(X)
3.54/1.73	   activate(n__s(X)) -> s(activate(X))
3.54/1.73	   activate(n__0) -> 0
3.54/1.73	   activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2))
3.54/1.73	   activate(n__fact(X)) -> fact(activate(X))
3.54/1.73	   activate(n__p(X)) -> p(activate(X))
3.54/1.73	   activate(X) -> X
3.54/1.73	
3.54/1.73	S is empty.
3.54/1.73	Rewrite Strategy: FULL
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(6) LowerBoundPropagationProof (FINISHED)
3.54/1.73	Propagated lower bound.
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(7)
3.54/1.73	BOUNDS(n^1, INF)
3.54/1.73	
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(8)
3.54/1.73	Obligation:
3.54/1.73	Analyzing the following TRS for decreasing loops:
3.54/1.73	
3.54/1.73	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.54/1.73	
3.54/1.73	
3.54/1.73	The TRS R consists of the following rules:
3.54/1.73	
3.54/1.73	   fact(X) -> if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X))))
3.54/1.73	   add(0, X) -> X
3.54/1.73	   add(s(X), Y) -> s(add(X, Y))
3.54/1.73	   prod(0, X) -> 0
3.54/1.73	   prod(s(X), Y) -> add(Y, prod(X, Y))
3.54/1.73	   if(true, X, Y) -> activate(X)
3.54/1.73	   if(false, X, Y) -> activate(Y)
3.54/1.73	   zero(0) -> true
3.54/1.73	   zero(s(X)) -> false
3.54/1.73	   p(s(X)) -> X
3.54/1.73	   s(X) -> n__s(X)
3.54/1.73	   0 -> n__0
3.54/1.73	   prod(X1, X2) -> n__prod(X1, X2)
3.54/1.73	   fact(X) -> n__fact(X)
3.54/1.73	   p(X) -> n__p(X)
3.54/1.73	   activate(n__s(X)) -> s(activate(X))
3.54/1.73	   activate(n__0) -> 0
3.54/1.73	   activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2))
3.54/1.73	   activate(n__fact(X)) -> fact(activate(X))
3.54/1.73	   activate(n__p(X)) -> p(activate(X))
3.54/1.73	   activate(X) -> X
3.54/1.73	
3.54/1.73	S is empty.
3.54/1.73	Rewrite Strategy: FULL
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(9) DecreasingLoopProof (FINISHED)
3.54/1.73	The following loop(s) give(s) rise to the lower bound EXP:
3.54/1.73	
3.54/1.73	The rewrite sequence
3.54/1.73	
3.54/1.73	activate(n__fact(X)) ->^+ if(zero(activate(X)), n__s(n__0), n__prod(activate(X), n__fact(n__p(activate(X)))))
3.54/1.73	
3.54/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.54/1.73	
3.54/1.73	The pumping substitution is [X / n__fact(X)].
3.54/1.73	
3.54/1.73	The result substitution is [ ].
3.54/1.73	
3.54/1.73	
3.54/1.73	
3.54/1.73	The rewrite sequence
3.54/1.73	
3.54/1.73	activate(n__fact(X)) ->^+ if(zero(activate(X)), n__s(n__0), n__prod(activate(X), n__fact(n__p(activate(X)))))
3.54/1.73	
3.54/1.73	gives rise to a decreasing loop by considering the right hand sides subterm at position [2,0].
3.54/1.73	
3.54/1.73	The pumping substitution is [X / n__fact(X)].
3.54/1.73	
3.54/1.73	The result substitution is [ ].
3.54/1.73	
3.54/1.73	
3.54/1.73	
3.54/1.73	
3.54/1.73	----------------------------------------
3.54/1.73	
3.54/1.73	(10)
3.54/1.73	BOUNDS(EXP, INF)
3.54/1.76	EOF