1126.57/291.53	WORST_CASE(Omega(n^1), ?)
1126.91/291.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1126.91/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.91/291.57	
1126.91/291.57	(0) CpxTRS
1126.91/291.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1126.91/291.57	(2) TRS for Loop Detection
1126.91/291.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1126.91/291.57	(4) BEST
1126.91/291.57	    (5) proven lower bound
1126.91/291.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1126.91/291.57	        (7) BOUNDS(n^1, INF)
1126.91/291.57	    (8) TRS for Loop Detection
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(0)
1126.91/291.57	Obligation:
1126.91/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	The TRS R consists of the following rules:
1126.91/291.57	
1126.91/291.57	   g(A) -> A
1126.91/291.57	   g(B) -> A
1126.91/291.57	   g(B) -> B
1126.91/291.57	   g(C) -> A
1126.91/291.57	   g(C) -> B
1126.91/291.57	   g(C) -> C
1126.91/291.57	   foldf(x, nil) -> x
1126.91/291.57	   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
1126.91/291.57	   f(t, x) -> f'(t, g(x))
1126.91/291.57	   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
1126.91/291.57	   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
1126.91/291.57	   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
1126.91/291.57	   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c)
1126.91/291.57	
1126.91/291.57	S is empty.
1126.91/291.57	Rewrite Strategy: INNERMOST
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1126.91/291.57	Transformed a relative TRS into a decreasing-loop problem.
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(2)
1126.91/291.57	Obligation:
1126.91/291.57	Analyzing the following TRS for decreasing loops:
1126.91/291.57	
1126.91/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	The TRS R consists of the following rules:
1126.91/291.57	
1126.91/291.57	   g(A) -> A
1126.91/291.57	   g(B) -> A
1126.91/291.57	   g(B) -> B
1126.91/291.57	   g(C) -> A
1126.91/291.57	   g(C) -> B
1126.91/291.57	   g(C) -> C
1126.91/291.57	   foldf(x, nil) -> x
1126.91/291.57	   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
1126.91/291.57	   f(t, x) -> f'(t, g(x))
1126.91/291.57	   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
1126.91/291.57	   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
1126.91/291.57	   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
1126.91/291.57	   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c)
1126.91/291.57	
1126.91/291.57	S is empty.
1126.91/291.57	Rewrite Strategy: INNERMOST
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
1126.91/291.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1126.91/291.57	
1126.91/291.57	The rewrite sequence
1126.91/291.57	
1126.91/291.57	foldf(x, cons(y, z)) ->^+ f(foldf(x, z), y)
1126.91/291.57	
1126.91/291.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1126.91/291.57	
1126.91/291.57	The pumping substitution is [z / cons(y, z)].
1126.91/291.57	
1126.91/291.57	The result substitution is [ ].
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(4)
1126.91/291.57	Complex Obligation (BEST)
1126.91/291.57	
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(5)
1126.91/291.57	Obligation:
1126.91/291.57	Proved the lower bound n^1 for the following obligation:
1126.91/291.57	
1126.91/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	The TRS R consists of the following rules:
1126.91/291.57	
1126.91/291.57	   g(A) -> A
1126.91/291.57	   g(B) -> A
1126.91/291.57	   g(B) -> B
1126.91/291.57	   g(C) -> A
1126.91/291.57	   g(C) -> B
1126.91/291.57	   g(C) -> C
1126.91/291.57	   foldf(x, nil) -> x
1126.91/291.57	   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
1126.91/291.57	   f(t, x) -> f'(t, g(x))
1126.91/291.57	   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
1126.91/291.57	   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
1126.91/291.57	   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
1126.91/291.57	   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c)
1126.91/291.57	
1126.91/291.57	S is empty.
1126.91/291.57	Rewrite Strategy: INNERMOST
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(6) LowerBoundPropagationProof (FINISHED)
1126.91/291.57	Propagated lower bound.
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(7)
1126.91/291.57	BOUNDS(n^1, INF)
1126.91/291.57	
1126.91/291.57	----------------------------------------
1126.91/291.57	
1126.91/291.57	(8)
1126.91/291.57	Obligation:
1126.91/291.57	Analyzing the following TRS for decreasing loops:
1126.91/291.57	
1126.91/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.91/291.57	
1126.91/291.57	
1126.91/291.57	The TRS R consists of the following rules:
1126.91/291.57	
1126.91/291.57	   g(A) -> A
1126.91/291.57	   g(B) -> A
1126.91/291.57	   g(B) -> B
1126.91/291.57	   g(C) -> A
1126.91/291.57	   g(C) -> B
1126.91/291.57	   g(C) -> C
1126.91/291.57	   foldf(x, nil) -> x
1126.91/291.57	   foldf(x, cons(y, z)) -> f(foldf(x, z), y)
1126.91/291.57	   f(t, x) -> f'(t, g(x))
1126.91/291.57	   f'(triple(a, b, c), C) -> triple(a, b, cons(C, c))
1126.91/291.57	   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
1126.91/291.57	   f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b))
1126.91/291.57	   f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c)
1126.91/291.57	
1126.91/291.57	S is empty.
1126.91/291.57	Rewrite Strategy: INNERMOST
1127.09/291.66	EOF