1121.71/291.50	WORST_CASE(Omega(n^1), ?)
1121.71/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1121.71/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.71/291.51	
1121.71/291.51	(0) CpxTRS
1121.71/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1121.71/291.51	(2) TRS for Loop Detection
1121.71/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1121.71/291.51	(4) BEST
1121.71/291.51	    (5) proven lower bound
1121.71/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1121.71/291.51	        (7) BOUNDS(n^1, INF)
1121.71/291.51	    (8) TRS for Loop Detection
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(0)
1121.71/291.51	Obligation:
1121.71/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	The TRS R consists of the following rules:
1121.71/291.51	
1121.71/291.51	   0(#) -> #
1121.71/291.51	   +(#, x) -> x
1121.71/291.51	   +(x, #) -> x
1121.71/291.51	   +(0(x), 0(y)) -> 0(+(x, y))
1121.71/291.51	   +(0(x), 1(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 0(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
1121.71/291.51	   +(+(x, y), z) -> +(x, +(y, z))
1121.71/291.51	   -(#, x) -> #
1121.71/291.51	   -(x, #) -> x
1121.71/291.51	   -(0(x), 0(y)) -> 0(-(x, y))
1121.71/291.51	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
1121.71/291.51	   -(1(x), 0(y)) -> 1(-(x, y))
1121.71/291.51	   -(1(x), 1(y)) -> 0(-(x, y))
1121.71/291.51	   not(true) -> false
1121.71/291.51	   not(false) -> true
1121.71/291.51	   if(true, x, y) -> x
1121.71/291.51	   if(false, x, y) -> y
1121.71/291.51	   ge(0(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(0(x), 1(y)) -> not(ge(y, x))
1121.71/291.51	   ge(1(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(1(x), 1(y)) -> ge(x, y)
1121.71/291.51	   ge(x, #) -> true
1121.71/291.51	   ge(#, 0(x)) -> ge(#, x)
1121.71/291.51	   ge(#, 1(x)) -> false
1121.71/291.51	   log(x) -> -(log'(x), 1(#))
1121.71/291.51	   log'(#) -> #
1121.71/291.51	   log'(1(x)) -> +(log'(x), 1(#))
1121.71/291.51	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
1121.71/291.51	
1121.71/291.51	S is empty.
1121.71/291.51	Rewrite Strategy: FULL
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1121.71/291.51	Transformed a relative TRS into a decreasing-loop problem.
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(2)
1121.71/291.51	Obligation:
1121.71/291.51	Analyzing the following TRS for decreasing loops:
1121.71/291.51	
1121.71/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	The TRS R consists of the following rules:
1121.71/291.51	
1121.71/291.51	   0(#) -> #
1121.71/291.51	   +(#, x) -> x
1121.71/291.51	   +(x, #) -> x
1121.71/291.51	   +(0(x), 0(y)) -> 0(+(x, y))
1121.71/291.51	   +(0(x), 1(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 0(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
1121.71/291.51	   +(+(x, y), z) -> +(x, +(y, z))
1121.71/291.51	   -(#, x) -> #
1121.71/291.51	   -(x, #) -> x
1121.71/291.51	   -(0(x), 0(y)) -> 0(-(x, y))
1121.71/291.51	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
1121.71/291.51	   -(1(x), 0(y)) -> 1(-(x, y))
1121.71/291.51	   -(1(x), 1(y)) -> 0(-(x, y))
1121.71/291.51	   not(true) -> false
1121.71/291.51	   not(false) -> true
1121.71/291.51	   if(true, x, y) -> x
1121.71/291.51	   if(false, x, y) -> y
1121.71/291.51	   ge(0(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(0(x), 1(y)) -> not(ge(y, x))
1121.71/291.51	   ge(1(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(1(x), 1(y)) -> ge(x, y)
1121.71/291.51	   ge(x, #) -> true
1121.71/291.51	   ge(#, 0(x)) -> ge(#, x)
1121.71/291.51	   ge(#, 1(x)) -> false
1121.71/291.51	   log(x) -> -(log'(x), 1(#))
1121.71/291.51	   log'(#) -> #
1121.71/291.51	   log'(1(x)) -> +(log'(x), 1(#))
1121.71/291.51	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
1121.71/291.51	
1121.71/291.51	S is empty.
1121.71/291.51	Rewrite Strategy: FULL
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
1121.71/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1121.71/291.51	
1121.71/291.51	The rewrite sequence
1121.71/291.51	
1121.71/291.51	+(1(x), 1(y)) ->^+ 0(+(+(x, y), 1(#)))
1121.71/291.51	
1121.71/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
1121.71/291.51	
1121.71/291.51	The pumping substitution is [x / 1(x), y / 1(y)].
1121.71/291.51	
1121.71/291.51	The result substitution is [ ].
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(4)
1121.71/291.51	Complex Obligation (BEST)
1121.71/291.51	
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(5)
1121.71/291.51	Obligation:
1121.71/291.51	Proved the lower bound n^1 for the following obligation:
1121.71/291.51	
1121.71/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	The TRS R consists of the following rules:
1121.71/291.51	
1121.71/291.51	   0(#) -> #
1121.71/291.51	   +(#, x) -> x
1121.71/291.51	   +(x, #) -> x
1121.71/291.51	   +(0(x), 0(y)) -> 0(+(x, y))
1121.71/291.51	   +(0(x), 1(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 0(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
1121.71/291.51	   +(+(x, y), z) -> +(x, +(y, z))
1121.71/291.51	   -(#, x) -> #
1121.71/291.51	   -(x, #) -> x
1121.71/291.51	   -(0(x), 0(y)) -> 0(-(x, y))
1121.71/291.51	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
1121.71/291.51	   -(1(x), 0(y)) -> 1(-(x, y))
1121.71/291.51	   -(1(x), 1(y)) -> 0(-(x, y))
1121.71/291.51	   not(true) -> false
1121.71/291.51	   not(false) -> true
1121.71/291.51	   if(true, x, y) -> x
1121.71/291.51	   if(false, x, y) -> y
1121.71/291.51	   ge(0(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(0(x), 1(y)) -> not(ge(y, x))
1121.71/291.51	   ge(1(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(1(x), 1(y)) -> ge(x, y)
1121.71/291.51	   ge(x, #) -> true
1121.71/291.51	   ge(#, 0(x)) -> ge(#, x)
1121.71/291.51	   ge(#, 1(x)) -> false
1121.71/291.51	   log(x) -> -(log'(x), 1(#))
1121.71/291.51	   log'(#) -> #
1121.71/291.51	   log'(1(x)) -> +(log'(x), 1(#))
1121.71/291.51	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
1121.71/291.51	
1121.71/291.51	S is empty.
1121.71/291.51	Rewrite Strategy: FULL
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(6) LowerBoundPropagationProof (FINISHED)
1121.71/291.51	Propagated lower bound.
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(7)
1121.71/291.51	BOUNDS(n^1, INF)
1121.71/291.51	
1121.71/291.51	----------------------------------------
1121.71/291.51	
1121.71/291.51	(8)
1121.71/291.51	Obligation:
1121.71/291.51	Analyzing the following TRS for decreasing loops:
1121.71/291.51	
1121.71/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.71/291.51	
1121.71/291.51	
1121.71/291.51	The TRS R consists of the following rules:
1121.71/291.51	
1121.71/291.51	   0(#) -> #
1121.71/291.51	   +(#, x) -> x
1121.71/291.51	   +(x, #) -> x
1121.71/291.51	   +(0(x), 0(y)) -> 0(+(x, y))
1121.71/291.51	   +(0(x), 1(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 0(y)) -> 1(+(x, y))
1121.71/291.51	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
1121.71/291.51	   +(+(x, y), z) -> +(x, +(y, z))
1121.71/291.51	   -(#, x) -> #
1121.71/291.51	   -(x, #) -> x
1121.71/291.51	   -(0(x), 0(y)) -> 0(-(x, y))
1121.71/291.51	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
1121.71/291.51	   -(1(x), 0(y)) -> 1(-(x, y))
1121.71/291.51	   -(1(x), 1(y)) -> 0(-(x, y))
1121.71/291.51	   not(true) -> false
1121.71/291.51	   not(false) -> true
1121.71/291.51	   if(true, x, y) -> x
1121.71/291.51	   if(false, x, y) -> y
1121.71/291.51	   ge(0(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(0(x), 1(y)) -> not(ge(y, x))
1121.71/291.51	   ge(1(x), 0(y)) -> ge(x, y)
1121.71/291.51	   ge(1(x), 1(y)) -> ge(x, y)
1121.71/291.51	   ge(x, #) -> true
1121.71/291.51	   ge(#, 0(x)) -> ge(#, x)
1121.71/291.51	   ge(#, 1(x)) -> false
1121.71/291.51	   log(x) -> -(log'(x), 1(#))
1121.71/291.51	   log'(#) -> #
1121.71/291.51	   log'(1(x)) -> +(log'(x), 1(#))
1121.71/291.51	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
1121.71/291.51	
1121.71/291.51	S is empty.
1121.71/291.51	Rewrite Strategy: FULL
1121.97/291.57	EOF