16.22/5.23	WORST_CASE(Omega(n^1), O(n^1))
16.22/5.24	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
16.22/5.24	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.22/5.24	
16.22/5.24	
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.22/5.24	
16.22/5.24	(0) CpxTRS
16.22/5.24	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
16.22/5.24	(2) CpxTRS
16.22/5.24	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
16.22/5.24	    (4) CpxTRS
16.22/5.24	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 18 ms]
16.22/5.24	    (6) BOUNDS(1, n^1)
16.22/5.24	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
16.22/5.24	(8) TRS for Loop Detection
16.22/5.24	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
16.22/5.24	    (10) BEST
16.22/5.24	        (11) proven lower bound
16.22/5.24	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
16.22/5.24	            (13) BOUNDS(n^1, INF)
16.22/5.24	        (14) TRS for Loop Detection
16.22/5.24	
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(0)
16.22/5.24	Obligation:
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.22/5.24	
16.22/5.24	
16.22/5.24	The TRS R consists of the following rules:
16.22/5.24	
16.22/5.24	   a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
16.22/5.24	   f(a(x), a(y)) -> a(f(x, y))
16.22/5.24	   f(b(x), b(y)) -> b(f(x, y))
16.22/5.24	
16.22/5.24	S is empty.
16.22/5.24	Rewrite Strategy: FULL
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
16.22/5.24	The TRS does not nest defined symbols.
16.22/5.24	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
16.22/5.24	a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
16.22/5.24	f(a(x), a(y)) -> a(f(x, y))
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(2)
16.22/5.24	Obligation:
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
16.22/5.24	
16.22/5.24	
16.22/5.24	The TRS R consists of the following rules:
16.22/5.24	
16.22/5.24	   f(b(x), b(y)) -> b(f(x, y))
16.22/5.24	
16.22/5.24	S is empty.
16.22/5.24	Rewrite Strategy: FULL
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(3) RelTrsToTrsProof (UPPER BOUND(ID))
16.22/5.24	transformed relative TRS to TRS
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(4)
16.22/5.24	Obligation:
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
16.22/5.24	
16.22/5.24	
16.22/5.24	The TRS R consists of the following rules:
16.22/5.24	
16.22/5.24	   f(b(x), b(y)) -> b(f(x, y))
16.22/5.24	
16.22/5.24	S is empty.
16.22/5.24	Rewrite Strategy: FULL
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(5) CpxTrsMatchBoundsTAProof (FINISHED)
16.22/5.24	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 
16.22/5.24	
16.22/5.24	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
16.22/5.24	final states : [1]
16.22/5.24	transitions: 
16.22/5.24	b0(0) -> 0
16.22/5.24	f0(0, 0) -> 1
16.22/5.24	f1(0, 0) -> 2
16.22/5.24	b1(2) -> 1
16.22/5.24	b1(2) -> 2
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(6)
16.22/5.24	BOUNDS(1, n^1)
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
16.22/5.24	Transformed a relative TRS into a decreasing-loop problem.
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(8)
16.22/5.24	Obligation:
16.22/5.24	Analyzing the following TRS for decreasing loops:
16.22/5.24	
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.22/5.24	
16.22/5.24	
16.22/5.24	The TRS R consists of the following rules:
16.22/5.24	
16.22/5.24	   a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
16.22/5.24	   f(a(x), a(y)) -> a(f(x, y))
16.22/5.24	   f(b(x), b(y)) -> b(f(x, y))
16.22/5.24	
16.22/5.24	S is empty.
16.22/5.24	Rewrite Strategy: FULL
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(9) DecreasingLoopProof (LOWER BOUND(ID))
16.22/5.24	The following loop(s) give(s) rise to the lower bound Omega(n^1):
16.22/5.24	
16.22/5.24	The rewrite sequence
16.22/5.24	
16.22/5.24	f(b(x), b(y)) ->^+ b(f(x, y))
16.22/5.24	
16.22/5.24	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
16.22/5.24	
16.22/5.24	The pumping substitution is [x / b(x), y / b(y)].
16.22/5.24	
16.22/5.24	The result substitution is [ ].
16.22/5.24	
16.22/5.24	
16.22/5.24	
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(10)
16.22/5.24	Complex Obligation (BEST)
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(11)
16.22/5.24	Obligation:
16.22/5.24	Proved the lower bound n^1 for the following obligation:
16.22/5.24	
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.22/5.24	
16.22/5.24	
16.22/5.24	The TRS R consists of the following rules:
16.22/5.24	
16.22/5.24	   a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
16.22/5.24	   f(a(x), a(y)) -> a(f(x, y))
16.22/5.24	   f(b(x), b(y)) -> b(f(x, y))
16.22/5.24	
16.22/5.24	S is empty.
16.22/5.24	Rewrite Strategy: FULL
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(12) LowerBoundPropagationProof (FINISHED)
16.22/5.24	Propagated lower bound.
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(13)
16.22/5.24	BOUNDS(n^1, INF)
16.22/5.24	
16.22/5.24	----------------------------------------
16.22/5.24	
16.22/5.24	(14)
16.22/5.24	Obligation:
16.22/5.24	Analyzing the following TRS for decreasing loops:
16.22/5.24	
16.22/5.24	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
16.22/5.24	
16.22/5.24	
16.22/5.24	The TRS R consists of the following rules:
16.22/5.24	
16.22/5.24	   a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
16.22/5.24	   f(a(x), a(y)) -> a(f(x, y))
16.22/5.24	   f(b(x), b(y)) -> b(f(x, y))
16.22/5.24	
16.22/5.24	S is empty.
16.22/5.24	Rewrite Strategy: FULL
16.34/5.28	EOF