3.11/1.61	WORST_CASE(NON_POLY, ?)
3.46/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.46/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.46/1.63	
3.46/1.63	
3.46/1.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.46/1.63	
3.46/1.63	(0) CpxTRS
3.46/1.63	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.46/1.63	(2) TRS for Loop Detection
3.46/1.63	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.46/1.63	(4) BEST
3.46/1.63	    (5) proven lower bound
3.46/1.63	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.46/1.63	        (7) BOUNDS(n^1, INF)
3.46/1.63	    (8) TRS for Loop Detection
3.46/1.63	        (9) DecreasingLoopProof [FINISHED, 17 ms]
3.46/1.63	        (10) BOUNDS(EXP, INF)
3.46/1.63	
3.46/1.63	
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(0)
3.46/1.63	Obligation:
3.46/1.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.46/1.63	
3.46/1.63	
3.46/1.63	The TRS R consists of the following rules:
3.46/1.63	
3.46/1.63	   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
3.46/1.63	   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
3.46/1.63	   h(0, S(x)) -> h(0, x)
3.46/1.63	   h(0, 0) -> 0
3.46/1.63	   g(S(x), 0) -> 0
3.46/1.63	   f(S(x), 0) -> 0
3.46/1.63	   h(S(x), x2) -> h(x, x2)
3.46/1.63	   g(0, x2) -> 0
3.46/1.63	   f(0, x2) -> 0
3.46/1.63	
3.46/1.63	S is empty.
3.46/1.63	Rewrite Strategy: INNERMOST
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.46/1.63	Transformed a relative TRS into a decreasing-loop problem.
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(2)
3.46/1.63	Obligation:
3.46/1.63	Analyzing the following TRS for decreasing loops:
3.46/1.63	
3.46/1.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.46/1.63	
3.46/1.63	
3.46/1.63	The TRS R consists of the following rules:
3.46/1.63	
3.46/1.63	   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
3.46/1.63	   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
3.46/1.63	   h(0, S(x)) -> h(0, x)
3.46/1.63	   h(0, 0) -> 0
3.46/1.63	   g(S(x), 0) -> 0
3.46/1.63	   f(S(x), 0) -> 0
3.46/1.63	   h(S(x), x2) -> h(x, x2)
3.46/1.63	   g(0, x2) -> 0
3.46/1.63	   f(0, x2) -> 0
3.46/1.63	
3.46/1.63	S is empty.
3.46/1.63	Rewrite Strategy: INNERMOST
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.46/1.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.46/1.63	
3.46/1.63	The rewrite sequence
3.46/1.63	
3.46/1.63	f(S(x'), S(x)) ->^+ h(g(x', S(x)), f(S(S(S(x'))), x))
3.46/1.63	
3.46/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.46/1.63	
3.46/1.63	The pumping substitution is [x / S(x)].
3.46/1.63	
3.46/1.63	The result substitution is [x' / S(S(x'))].
3.46/1.63	
3.46/1.63	
3.46/1.63	
3.46/1.63	
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(4)
3.46/1.63	Complex Obligation (BEST)
3.46/1.63	
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(5)
3.46/1.63	Obligation:
3.46/1.63	Proved the lower bound n^1 for the following obligation:
3.46/1.63	
3.46/1.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.46/1.63	
3.46/1.63	
3.46/1.63	The TRS R consists of the following rules:
3.46/1.63	
3.46/1.63	   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
3.46/1.63	   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
3.46/1.63	   h(0, S(x)) -> h(0, x)
3.46/1.63	   h(0, 0) -> 0
3.46/1.63	   g(S(x), 0) -> 0
3.46/1.63	   f(S(x), 0) -> 0
3.46/1.63	   h(S(x), x2) -> h(x, x2)
3.46/1.63	   g(0, x2) -> 0
3.46/1.63	   f(0, x2) -> 0
3.46/1.63	
3.46/1.63	S is empty.
3.46/1.63	Rewrite Strategy: INNERMOST
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(6) LowerBoundPropagationProof (FINISHED)
3.46/1.63	Propagated lower bound.
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(7)
3.46/1.63	BOUNDS(n^1, INF)
3.46/1.63	
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(8)
3.46/1.63	Obligation:
3.46/1.63	Analyzing the following TRS for decreasing loops:
3.46/1.63	
3.46/1.63	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.46/1.63	
3.46/1.63	
3.46/1.63	The TRS R consists of the following rules:
3.46/1.63	
3.46/1.63	   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
3.46/1.63	   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
3.46/1.63	   h(0, S(x)) -> h(0, x)
3.46/1.63	   h(0, 0) -> 0
3.46/1.63	   g(S(x), 0) -> 0
3.46/1.63	   f(S(x), 0) -> 0
3.46/1.63	   h(S(x), x2) -> h(x, x2)
3.46/1.63	   g(0, x2) -> 0
3.46/1.63	   f(0, x2) -> 0
3.46/1.63	
3.46/1.63	S is empty.
3.46/1.63	Rewrite Strategy: INNERMOST
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(9) DecreasingLoopProof (FINISHED)
3.46/1.63	The following loop(s) give(s) rise to the lower bound EXP:
3.46/1.63	
3.46/1.63	The rewrite sequence
3.46/1.63	
3.46/1.63	g(S(x), S(x')) ->^+ h(h(g(x, S(x')), f(S(S(S(x))), x')), g(x, S(S(S(x')))))
3.46/1.63	
3.46/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.46/1.63	
3.46/1.63	The pumping substitution is [x / S(x)].
3.46/1.63	
3.46/1.63	The result substitution is [ ].
3.46/1.63	
3.46/1.63	
3.46/1.63	
3.46/1.63	The rewrite sequence
3.46/1.63	
3.46/1.63	g(S(x), S(x')) ->^+ h(h(g(x, S(x')), f(S(S(S(x))), x')), g(x, S(S(S(x')))))
3.46/1.63	
3.46/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.46/1.63	
3.46/1.63	The pumping substitution is [x / S(x)].
3.46/1.63	
3.46/1.63	The result substitution is [x' / S(S(x'))].
3.46/1.63	
3.46/1.63	
3.46/1.63	
3.46/1.63	
3.46/1.63	----------------------------------------
3.46/1.63	
3.46/1.63	(10)
3.46/1.63	BOUNDS(EXP, INF)
3.46/1.65	EOF