1113.26/291.49	WORST_CASE(Omega(n^1), ?)
1113.54/291.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1113.54/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.54/291.50	
1113.54/291.50	(0) CpxTRS
1113.54/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1113.54/291.50	(2) TRS for Loop Detection
1113.54/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1113.54/291.50	(4) BEST
1113.54/291.50	    (5) proven lower bound
1113.54/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1113.54/291.50	        (7) BOUNDS(n^1, INF)
1113.54/291.50	    (8) TRS for Loop Detection
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(0)
1113.54/291.50	Obligation:
1113.54/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	The TRS R consists of the following rules:
1113.54/291.50	
1113.54/291.50	   le(0, y) -> true
1113.54/291.50	   le(s(x), 0) -> false
1113.54/291.50	   le(s(x), s(y)) -> le(x, y)
1113.54/291.50	   eq(0, 0) -> true
1113.54/291.50	   eq(0, s(y)) -> false
1113.54/291.50	   eq(s(x), 0) -> false
1113.54/291.50	   eq(s(x), s(y)) -> eq(x, y)
1113.54/291.50	   minsort(nil) -> nil
1113.54/291.50	   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
1113.54/291.50	   min(nil) -> 0
1113.54/291.50	   min(cons(x, nil)) -> x
1113.54/291.50	   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
1113.54/291.50	   if1(true, x, y, xs) -> min(cons(x, xs))
1113.54/291.50	   if1(false, x, y, xs) -> min(cons(y, xs))
1113.54/291.50	   rm(x, nil) -> nil
1113.54/291.50	   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
1113.54/291.50	   if2(true, x, y, xs) -> rm(x, xs)
1113.54/291.50	   if2(false, x, y, xs) -> cons(y, rm(x, xs))
1113.54/291.50	
1113.54/291.50	S is empty.
1113.54/291.50	Rewrite Strategy: INNERMOST
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1113.54/291.50	Transformed a relative TRS into a decreasing-loop problem.
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(2)
1113.54/291.50	Obligation:
1113.54/291.50	Analyzing the following TRS for decreasing loops:
1113.54/291.50	
1113.54/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	The TRS R consists of the following rules:
1113.54/291.50	
1113.54/291.50	   le(0, y) -> true
1113.54/291.50	   le(s(x), 0) -> false
1113.54/291.50	   le(s(x), s(y)) -> le(x, y)
1113.54/291.50	   eq(0, 0) -> true
1113.54/291.50	   eq(0, s(y)) -> false
1113.54/291.50	   eq(s(x), 0) -> false
1113.54/291.50	   eq(s(x), s(y)) -> eq(x, y)
1113.54/291.50	   minsort(nil) -> nil
1113.54/291.50	   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
1113.54/291.50	   min(nil) -> 0
1113.54/291.50	   min(cons(x, nil)) -> x
1113.54/291.50	   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
1113.54/291.50	   if1(true, x, y, xs) -> min(cons(x, xs))
1113.54/291.50	   if1(false, x, y, xs) -> min(cons(y, xs))
1113.54/291.50	   rm(x, nil) -> nil
1113.54/291.50	   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
1113.54/291.50	   if2(true, x, y, xs) -> rm(x, xs)
1113.54/291.50	   if2(false, x, y, xs) -> cons(y, rm(x, xs))
1113.54/291.50	
1113.54/291.50	S is empty.
1113.54/291.50	Rewrite Strategy: INNERMOST
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
1113.54/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1113.54/291.50	
1113.54/291.50	The rewrite sequence
1113.54/291.50	
1113.54/291.50	le(s(x), s(y)) ->^+ le(x, y)
1113.54/291.50	
1113.54/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1113.54/291.50	
1113.54/291.50	The pumping substitution is [x / s(x), y / s(y)].
1113.54/291.50	
1113.54/291.50	The result substitution is [ ].
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(4)
1113.54/291.50	Complex Obligation (BEST)
1113.54/291.50	
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(5)
1113.54/291.50	Obligation:
1113.54/291.50	Proved the lower bound n^1 for the following obligation:
1113.54/291.50	
1113.54/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	The TRS R consists of the following rules:
1113.54/291.50	
1113.54/291.50	   le(0, y) -> true
1113.54/291.50	   le(s(x), 0) -> false
1113.54/291.50	   le(s(x), s(y)) -> le(x, y)
1113.54/291.50	   eq(0, 0) -> true
1113.54/291.50	   eq(0, s(y)) -> false
1113.54/291.50	   eq(s(x), 0) -> false
1113.54/291.50	   eq(s(x), s(y)) -> eq(x, y)
1113.54/291.50	   minsort(nil) -> nil
1113.54/291.50	   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
1113.54/291.50	   min(nil) -> 0
1113.54/291.50	   min(cons(x, nil)) -> x
1113.54/291.50	   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
1113.54/291.50	   if1(true, x, y, xs) -> min(cons(x, xs))
1113.54/291.50	   if1(false, x, y, xs) -> min(cons(y, xs))
1113.54/291.50	   rm(x, nil) -> nil
1113.54/291.50	   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
1113.54/291.50	   if2(true, x, y, xs) -> rm(x, xs)
1113.54/291.50	   if2(false, x, y, xs) -> cons(y, rm(x, xs))
1113.54/291.50	
1113.54/291.50	S is empty.
1113.54/291.50	Rewrite Strategy: INNERMOST
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(6) LowerBoundPropagationProof (FINISHED)
1113.54/291.50	Propagated lower bound.
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(7)
1113.54/291.50	BOUNDS(n^1, INF)
1113.54/291.50	
1113.54/291.50	----------------------------------------
1113.54/291.50	
1113.54/291.50	(8)
1113.54/291.50	Obligation:
1113.54/291.50	Analyzing the following TRS for decreasing loops:
1113.54/291.50	
1113.54/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.54/291.50	
1113.54/291.50	
1113.54/291.50	The TRS R consists of the following rules:
1113.54/291.50	
1113.54/291.50	   le(0, y) -> true
1113.54/291.50	   le(s(x), 0) -> false
1113.54/291.50	   le(s(x), s(y)) -> le(x, y)
1113.54/291.50	   eq(0, 0) -> true
1113.54/291.50	   eq(0, s(y)) -> false
1113.54/291.50	   eq(s(x), 0) -> false
1113.54/291.50	   eq(s(x), s(y)) -> eq(x, y)
1113.54/291.50	   minsort(nil) -> nil
1113.54/291.50	   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
1113.54/291.50	   min(nil) -> 0
1113.54/291.50	   min(cons(x, nil)) -> x
1113.54/291.50	   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
1113.54/291.50	   if1(true, x, y, xs) -> min(cons(x, xs))
1113.54/291.50	   if1(false, x, y, xs) -> min(cons(y, xs))
1113.54/291.50	   rm(x, nil) -> nil
1113.54/291.50	   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
1113.54/291.50	   if2(true, x, y, xs) -> rm(x, xs)
1113.54/291.50	   if2(false, x, y, xs) -> cons(y, rm(x, xs))
1113.54/291.50	
1113.54/291.50	S is empty.
1113.54/291.50	Rewrite Strategy: INNERMOST
1113.67/291.58	EOF