1004.54/291.49	WORST_CASE(Omega(n^1), ?)
1004.54/291.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1004.54/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1004.54/291.49	
1004.54/291.49	(0) CpxTRS
1004.54/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1004.54/291.49	(2) TRS for Loop Detection
1004.54/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1004.54/291.49	(4) BEST
1004.54/291.49	    (5) proven lower bound
1004.54/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1004.54/291.49	        (7) BOUNDS(n^1, INF)
1004.54/291.49	    (8) TRS for Loop Detection
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(0)
1004.54/291.49	Obligation:
1004.54/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	The TRS R consists of the following rules:
1004.54/291.49	
1004.54/291.49	   function(iszero, 0, dummy, dummy2) -> true
1004.54/291.49	   function(iszero, s(x), dummy, dummy2) -> false
1004.54/291.49	   function(p, 0, dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(0), dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
1004.54/291.49	   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
1004.54/291.49	   function(if, true, x, y) -> y
1004.54/291.49	   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
1004.54/291.49	   function(third, x, y, z) -> z
1004.54/291.49	
1004.54/291.49	S is empty.
1004.54/291.49	Rewrite Strategy: INNERMOST
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1004.54/291.49	Transformed a relative TRS into a decreasing-loop problem.
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(2)
1004.54/291.49	Obligation:
1004.54/291.49	Analyzing the following TRS for decreasing loops:
1004.54/291.49	
1004.54/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	The TRS R consists of the following rules:
1004.54/291.49	
1004.54/291.49	   function(iszero, 0, dummy, dummy2) -> true
1004.54/291.49	   function(iszero, s(x), dummy, dummy2) -> false
1004.54/291.49	   function(p, 0, dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(0), dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
1004.54/291.49	   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
1004.54/291.49	   function(if, true, x, y) -> y
1004.54/291.49	   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
1004.54/291.49	   function(third, x, y, z) -> z
1004.54/291.49	
1004.54/291.49	S is empty.
1004.54/291.49	Rewrite Strategy: INNERMOST
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
1004.54/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1004.54/291.49	
1004.54/291.49	The rewrite sequence
1004.54/291.49	
1004.54/291.49	function(p, s(s(x)), dummy, dummy2) ->^+ s(function(p, s(x), x, x))
1004.54/291.49	
1004.54/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1004.54/291.49	
1004.54/291.49	The pumping substitution is [x / s(x)].
1004.54/291.49	
1004.54/291.49	The result substitution is [dummy / x, dummy2 / x].
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(4)
1004.54/291.49	Complex Obligation (BEST)
1004.54/291.49	
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(5)
1004.54/291.49	Obligation:
1004.54/291.49	Proved the lower bound n^1 for the following obligation:
1004.54/291.49	
1004.54/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	The TRS R consists of the following rules:
1004.54/291.49	
1004.54/291.49	   function(iszero, 0, dummy, dummy2) -> true
1004.54/291.49	   function(iszero, s(x), dummy, dummy2) -> false
1004.54/291.49	   function(p, 0, dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(0), dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
1004.54/291.49	   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
1004.54/291.49	   function(if, true, x, y) -> y
1004.54/291.49	   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
1004.54/291.49	   function(third, x, y, z) -> z
1004.54/291.49	
1004.54/291.49	S is empty.
1004.54/291.49	Rewrite Strategy: INNERMOST
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(6) LowerBoundPropagationProof (FINISHED)
1004.54/291.49	Propagated lower bound.
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(7)
1004.54/291.49	BOUNDS(n^1, INF)
1004.54/291.49	
1004.54/291.49	----------------------------------------
1004.54/291.49	
1004.54/291.49	(8)
1004.54/291.49	Obligation:
1004.54/291.49	Analyzing the following TRS for decreasing loops:
1004.54/291.49	
1004.54/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1004.54/291.49	
1004.54/291.49	
1004.54/291.49	The TRS R consists of the following rules:
1004.54/291.49	
1004.54/291.49	   function(iszero, 0, dummy, dummy2) -> true
1004.54/291.49	   function(iszero, s(x), dummy, dummy2) -> false
1004.54/291.49	   function(p, 0, dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(0), dummy, dummy2) -> 0
1004.54/291.49	   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
1004.54/291.49	   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
1004.54/291.49	   function(if, true, x, y) -> y
1004.54/291.49	   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
1004.54/291.49	   function(third, x, y, z) -> z
1004.54/291.49	
1004.54/291.49	S is empty.
1004.54/291.49	Rewrite Strategy: INNERMOST
1004.65/291.56	EOF