1113.31/291.83	WORST_CASE(Omega(n^1), ?)
1113.37/291.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1113.37/291.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.37/291.87	
1113.37/291.87	(0) CpxTRS
1113.37/291.87	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1113.37/291.87	(2) TRS for Loop Detection
1113.37/291.87	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1113.37/291.87	(4) BEST
1113.37/291.87	    (5) proven lower bound
1113.37/291.87	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1113.37/291.87	        (7) BOUNDS(n^1, INF)
1113.37/291.87	    (8) TRS for Loop Detection
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(0)
1113.37/291.87	Obligation:
1113.37/291.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	The TRS R consists of the following rules:
1113.37/291.87	
1113.37/291.87	   eq(0, 0) -> true
1113.37/291.87	   eq(0, s(x)) -> false
1113.37/291.87	   eq(s(x), 0) -> false
1113.37/291.87	   eq(s(x), s(y)) -> eq(x, y)
1113.37/291.87	   le(0, y) -> true
1113.37/291.87	   le(s(x), 0) -> false
1113.37/291.87	   le(s(x), s(y)) -> le(x, y)
1113.37/291.87	   app(nil, y) -> y
1113.37/291.87	   app(add(n, x), y) -> add(n, app(x, y))
1113.37/291.87	   min(nil) -> 0
1113.37/291.87	   min(add(n, x)) -> minIter(add(n, x), add(n, x), 0)
1113.37/291.87	   minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m))
1113.37/291.87	   minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m)
1113.37/291.87	   if_min(true, x, y, m) -> m
1113.37/291.87	   if_min(false, x, y, m) -> minIter(x, y, m)
1113.37/291.87	   head(add(n, x)) -> n
1113.37/291.87	   tail(add(n, x)) -> x
1113.37/291.87	   tail(nil) -> nil
1113.37/291.87	   null(nil) -> true
1113.37/291.87	   null(add(n, x)) -> false
1113.37/291.87	   rm(n, nil) -> nil
1113.37/291.87	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
1113.37/291.87	   if_rm(true, n, add(m, x)) -> rm(n, x)
1113.37/291.87	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
1113.37/291.87	   minsort(nil, nil) -> nil
1113.37/291.87	   minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
1113.37/291.87	   if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
1113.37/291.87	   if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))
1113.37/291.87	
1113.37/291.87	S is empty.
1113.37/291.87	Rewrite Strategy: INNERMOST
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1113.37/291.87	Transformed a relative TRS into a decreasing-loop problem.
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(2)
1113.37/291.87	Obligation:
1113.37/291.87	Analyzing the following TRS for decreasing loops:
1113.37/291.87	
1113.37/291.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	The TRS R consists of the following rules:
1113.37/291.87	
1113.37/291.87	   eq(0, 0) -> true
1113.37/291.87	   eq(0, s(x)) -> false
1113.37/291.87	   eq(s(x), 0) -> false
1113.37/291.87	   eq(s(x), s(y)) -> eq(x, y)
1113.37/291.87	   le(0, y) -> true
1113.37/291.87	   le(s(x), 0) -> false
1113.37/291.87	   le(s(x), s(y)) -> le(x, y)
1113.37/291.87	   app(nil, y) -> y
1113.37/291.87	   app(add(n, x), y) -> add(n, app(x, y))
1113.37/291.87	   min(nil) -> 0
1113.37/291.87	   min(add(n, x)) -> minIter(add(n, x), add(n, x), 0)
1113.37/291.87	   minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m))
1113.37/291.87	   minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m)
1113.37/291.87	   if_min(true, x, y, m) -> m
1113.37/291.87	   if_min(false, x, y, m) -> minIter(x, y, m)
1113.37/291.87	   head(add(n, x)) -> n
1113.37/291.87	   tail(add(n, x)) -> x
1113.37/291.87	   tail(nil) -> nil
1113.37/291.87	   null(nil) -> true
1113.37/291.87	   null(add(n, x)) -> false
1113.37/291.87	   rm(n, nil) -> nil
1113.37/291.87	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
1113.37/291.87	   if_rm(true, n, add(m, x)) -> rm(n, x)
1113.37/291.87	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
1113.37/291.87	   minsort(nil, nil) -> nil
1113.37/291.87	   minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
1113.37/291.87	   if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
1113.37/291.87	   if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))
1113.37/291.87	
1113.37/291.87	S is empty.
1113.37/291.87	Rewrite Strategy: INNERMOST
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(3) DecreasingLoopProof (LOWER BOUND(ID))
1113.37/291.87	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1113.37/291.87	
1113.37/291.87	The rewrite sequence
1113.37/291.87	
1113.37/291.87	le(s(x), s(y)) ->^+ le(x, y)
1113.37/291.87	
1113.37/291.87	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1113.37/291.87	
1113.37/291.87	The pumping substitution is [x / s(x), y / s(y)].
1113.37/291.87	
1113.37/291.87	The result substitution is [ ].
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(4)
1113.37/291.87	Complex Obligation (BEST)
1113.37/291.87	
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(5)
1113.37/291.87	Obligation:
1113.37/291.87	Proved the lower bound n^1 for the following obligation:
1113.37/291.87	
1113.37/291.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	The TRS R consists of the following rules:
1113.37/291.87	
1113.37/291.87	   eq(0, 0) -> true
1113.37/291.87	   eq(0, s(x)) -> false
1113.37/291.87	   eq(s(x), 0) -> false
1113.37/291.87	   eq(s(x), s(y)) -> eq(x, y)
1113.37/291.87	   le(0, y) -> true
1113.37/291.87	   le(s(x), 0) -> false
1113.37/291.87	   le(s(x), s(y)) -> le(x, y)
1113.37/291.87	   app(nil, y) -> y
1113.37/291.87	   app(add(n, x), y) -> add(n, app(x, y))
1113.37/291.87	   min(nil) -> 0
1113.37/291.87	   min(add(n, x)) -> minIter(add(n, x), add(n, x), 0)
1113.37/291.87	   minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m))
1113.37/291.87	   minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m)
1113.37/291.87	   if_min(true, x, y, m) -> m
1113.37/291.87	   if_min(false, x, y, m) -> minIter(x, y, m)
1113.37/291.87	   head(add(n, x)) -> n
1113.37/291.87	   tail(add(n, x)) -> x
1113.37/291.87	   tail(nil) -> nil
1113.37/291.87	   null(nil) -> true
1113.37/291.87	   null(add(n, x)) -> false
1113.37/291.87	   rm(n, nil) -> nil
1113.37/291.87	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
1113.37/291.87	   if_rm(true, n, add(m, x)) -> rm(n, x)
1113.37/291.87	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
1113.37/291.87	   minsort(nil, nil) -> nil
1113.37/291.87	   minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
1113.37/291.87	   if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
1113.37/291.87	   if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))
1113.37/291.87	
1113.37/291.87	S is empty.
1113.37/291.87	Rewrite Strategy: INNERMOST
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(6) LowerBoundPropagationProof (FINISHED)
1113.37/291.87	Propagated lower bound.
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(7)
1113.37/291.87	BOUNDS(n^1, INF)
1113.37/291.87	
1113.37/291.87	----------------------------------------
1113.37/291.87	
1113.37/291.87	(8)
1113.37/291.87	Obligation:
1113.37/291.87	Analyzing the following TRS for decreasing loops:
1113.37/291.87	
1113.37/291.87	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1113.37/291.87	
1113.37/291.87	
1113.37/291.87	The TRS R consists of the following rules:
1113.37/291.87	
1113.37/291.87	   eq(0, 0) -> true
1113.37/291.87	   eq(0, s(x)) -> false
1113.37/291.87	   eq(s(x), 0) -> false
1113.37/291.87	   eq(s(x), s(y)) -> eq(x, y)
1113.37/291.87	   le(0, y) -> true
1113.37/291.87	   le(s(x), 0) -> false
1113.37/291.87	   le(s(x), s(y)) -> le(x, y)
1113.37/291.87	   app(nil, y) -> y
1113.37/291.87	   app(add(n, x), y) -> add(n, app(x, y))
1113.37/291.87	   min(nil) -> 0
1113.37/291.87	   min(add(n, x)) -> minIter(add(n, x), add(n, x), 0)
1113.37/291.87	   minIter(nil, add(n, y), m) -> minIter(add(n, y), add(n, y), s(m))
1113.37/291.87	   minIter(add(n, x), y, m) -> if_min(le(n, m), x, y, m)
1113.37/291.87	   if_min(true, x, y, m) -> m
1113.37/291.87	   if_min(false, x, y, m) -> minIter(x, y, m)
1113.37/291.87	   head(add(n, x)) -> n
1113.37/291.87	   tail(add(n, x)) -> x
1113.37/291.87	   tail(nil) -> nil
1113.37/291.87	   null(nil) -> true
1113.37/291.87	   null(add(n, x)) -> false
1113.37/291.87	   rm(n, nil) -> nil
1113.37/291.87	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
1113.37/291.87	   if_rm(true, n, add(m, x)) -> rm(n, x)
1113.37/291.87	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
1113.37/291.87	   minsort(nil, nil) -> nil
1113.37/291.87	   minsort(add(n, x), y) -> if_minsort(eq(n, min(add(n, x))), add(n, x), y)
1113.37/291.87	   if_minsort(true, add(n, x), y) -> add(n, minsort(app(rm(n, x), y), nil))
1113.37/291.87	   if_minsort(false, add(n, x), y) -> minsort(x, add(n, y))
1113.37/291.87	
1113.37/291.87	S is empty.
1113.37/291.87	Rewrite Strategy: INNERMOST
1113.58/291.96	EOF