3.70/1.75	WORST_CASE(NON_POLY, ?)
3.70/1.76	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.70/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.70/1.76	
3.70/1.76	
3.70/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.70/1.76	
3.70/1.76	(0) CpxTRS
3.70/1.76	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.70/1.76	(2) TRS for Loop Detection
3.70/1.76	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.70/1.76	(4) BEST
3.70/1.76	    (5) proven lower bound
3.70/1.76	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.70/1.76	        (7) BOUNDS(n^1, INF)
3.70/1.76	    (8) TRS for Loop Detection
3.70/1.76	        (9) DecreasingLoopProof [FINISHED, 108 ms]
3.70/1.76	        (10) BOUNDS(EXP, INF)
3.70/1.76	
3.70/1.76	
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(0)
3.70/1.76	Obligation:
3.70/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.70/1.76	
3.70/1.76	
3.70/1.76	The TRS R consists of the following rules:
3.70/1.76	
3.70/1.76	   minus(n__0, Y) -> 0
3.70/1.76	   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
3.70/1.76	   geq(X, n__0) -> true
3.70/1.76	   geq(n__0, n__s(Y)) -> false
3.70/1.76	   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
3.70/1.76	   div(0, n__s(Y)) -> 0
3.70/1.76	   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
3.70/1.76	   if(true, X, Y) -> activate(X)
3.70/1.76	   if(false, X, Y) -> activate(Y)
3.70/1.76	   0 -> n__0
3.70/1.76	   s(X) -> n__s(X)
3.70/1.76	   div(X1, X2) -> n__div(X1, X2)
3.70/1.76	   minus(X1, X2) -> n__minus(X1, X2)
3.70/1.76	   activate(n__0) -> 0
3.70/1.76	   activate(n__s(X)) -> s(activate(X))
3.70/1.76	   activate(n__div(X1, X2)) -> div(activate(X1), X2)
3.70/1.76	   activate(n__minus(X1, X2)) -> minus(X1, X2)
3.70/1.76	   activate(X) -> X
3.70/1.76	
3.70/1.76	S is empty.
3.70/1.76	Rewrite Strategy: FULL
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.70/1.76	Transformed a relative TRS into a decreasing-loop problem.
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(2)
3.70/1.76	Obligation:
3.70/1.76	Analyzing the following TRS for decreasing loops:
3.70/1.76	
3.70/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.70/1.76	
3.70/1.76	
3.70/1.76	The TRS R consists of the following rules:
3.70/1.76	
3.70/1.76	   minus(n__0, Y) -> 0
3.70/1.76	   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
3.70/1.76	   geq(X, n__0) -> true
3.70/1.76	   geq(n__0, n__s(Y)) -> false
3.70/1.76	   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
3.70/1.76	   div(0, n__s(Y)) -> 0
3.70/1.76	   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
3.70/1.76	   if(true, X, Y) -> activate(X)
3.70/1.76	   if(false, X, Y) -> activate(Y)
3.70/1.76	   0 -> n__0
3.70/1.76	   s(X) -> n__s(X)
3.70/1.76	   div(X1, X2) -> n__div(X1, X2)
3.70/1.76	   minus(X1, X2) -> n__minus(X1, X2)
3.70/1.76	   activate(n__0) -> 0
3.70/1.76	   activate(n__s(X)) -> s(activate(X))
3.70/1.76	   activate(n__div(X1, X2)) -> div(activate(X1), X2)
3.70/1.76	   activate(n__minus(X1, X2)) -> minus(X1, X2)
3.70/1.76	   activate(X) -> X
3.70/1.76	
3.70/1.76	S is empty.
3.70/1.76	Rewrite Strategy: FULL
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.70/1.76	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.70/1.76	
3.70/1.76	The rewrite sequence
3.70/1.76	
3.70/1.76	activate(n__s(X)) ->^+ s(activate(X))
3.70/1.76	
3.70/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.70/1.76	
3.70/1.76	The pumping substitution is [X / n__s(X)].
3.70/1.76	
3.70/1.76	The result substitution is [ ].
3.70/1.76	
3.70/1.76	
3.70/1.76	
3.70/1.76	
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(4)
3.70/1.76	Complex Obligation (BEST)
3.70/1.76	
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(5)
3.70/1.76	Obligation:
3.70/1.76	Proved the lower bound n^1 for the following obligation:
3.70/1.76	
3.70/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.70/1.76	
3.70/1.76	
3.70/1.76	The TRS R consists of the following rules:
3.70/1.76	
3.70/1.76	   minus(n__0, Y) -> 0
3.70/1.76	   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
3.70/1.76	   geq(X, n__0) -> true
3.70/1.76	   geq(n__0, n__s(Y)) -> false
3.70/1.76	   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
3.70/1.76	   div(0, n__s(Y)) -> 0
3.70/1.76	   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
3.70/1.76	   if(true, X, Y) -> activate(X)
3.70/1.76	   if(false, X, Y) -> activate(Y)
3.70/1.76	   0 -> n__0
3.70/1.76	   s(X) -> n__s(X)
3.70/1.76	   div(X1, X2) -> n__div(X1, X2)
3.70/1.76	   minus(X1, X2) -> n__minus(X1, X2)
3.70/1.76	   activate(n__0) -> 0
3.70/1.76	   activate(n__s(X)) -> s(activate(X))
3.70/1.76	   activate(n__div(X1, X2)) -> div(activate(X1), X2)
3.70/1.76	   activate(n__minus(X1, X2)) -> minus(X1, X2)
3.70/1.76	   activate(X) -> X
3.70/1.76	
3.70/1.76	S is empty.
3.70/1.76	Rewrite Strategy: FULL
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(6) LowerBoundPropagationProof (FINISHED)
3.70/1.76	Propagated lower bound.
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(7)
3.70/1.76	BOUNDS(n^1, INF)
3.70/1.76	
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(8)
3.70/1.76	Obligation:
3.70/1.76	Analyzing the following TRS for decreasing loops:
3.70/1.76	
3.70/1.76	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.70/1.76	
3.70/1.76	
3.70/1.76	The TRS R consists of the following rules:
3.70/1.76	
3.70/1.76	   minus(n__0, Y) -> 0
3.70/1.76	   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
3.70/1.76	   geq(X, n__0) -> true
3.70/1.76	   geq(n__0, n__s(Y)) -> false
3.70/1.76	   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
3.70/1.76	   div(0, n__s(Y)) -> 0
3.70/1.76	   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
3.70/1.76	   if(true, X, Y) -> activate(X)
3.70/1.76	   if(false, X, Y) -> activate(Y)
3.70/1.76	   0 -> n__0
3.70/1.76	   s(X) -> n__s(X)
3.70/1.76	   div(X1, X2) -> n__div(X1, X2)
3.70/1.76	   minus(X1, X2) -> n__minus(X1, X2)
3.70/1.76	   activate(n__0) -> 0
3.70/1.76	   activate(n__s(X)) -> s(activate(X))
3.70/1.76	   activate(n__div(X1, X2)) -> div(activate(X1), X2)
3.70/1.76	   activate(n__minus(X1, X2)) -> minus(X1, X2)
3.70/1.76	   activate(X) -> X
3.70/1.76	
3.70/1.76	S is empty.
3.70/1.76	Rewrite Strategy: FULL
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(9) DecreasingLoopProof (FINISHED)
3.70/1.76	The following loop(s) give(s) rise to the lower bound EXP:
3.70/1.76	
3.70/1.76	The rewrite sequence
3.70/1.76	
3.70/1.76	activate(n__div(n__s(X1_0), n__s(Y2_1))) ->^+ if(geq(activate(X1_0), activate(Y2_1)), n__s(n__div(n__minus(activate(X1_0), activate(Y2_1)), n__s(activate(Y2_1)))), n__0)
3.70/1.76	
3.70/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.70/1.76	
3.70/1.76	The pumping substitution is [X1_0 / n__div(n__s(X1_0), n__s(Y2_1))].
3.70/1.76	
3.70/1.76	The result substitution is [ ].
3.70/1.76	
3.70/1.76	
3.70/1.76	
3.70/1.76	The rewrite sequence
3.70/1.76	
3.70/1.76	activate(n__div(n__s(X1_0), n__s(Y2_1))) ->^+ if(geq(activate(X1_0), activate(Y2_1)), n__s(n__div(n__minus(activate(X1_0), activate(Y2_1)), n__s(activate(Y2_1)))), n__0)
3.70/1.76	
3.70/1.76	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0,0].
3.70/1.76	
3.70/1.76	The pumping substitution is [X1_0 / n__div(n__s(X1_0), n__s(Y2_1))].
3.70/1.76	
3.70/1.76	The result substitution is [ ].
3.70/1.76	
3.70/1.76	
3.70/1.76	
3.70/1.76	
3.70/1.76	----------------------------------------
3.70/1.76	
3.70/1.76	(10)
3.70/1.76	BOUNDS(EXP, INF)
4.09/1.84	EOF