308.55/291.61	WORST_CASE(Omega(n^1), ?)
308.55/291.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
308.55/291.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
308.55/291.62	
308.55/291.62	
308.55/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.55/291.62	
308.55/291.62	(0) CpxTRS
308.55/291.62	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
308.55/291.62	(2) TRS for Loop Detection
308.55/291.62	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
308.55/291.62	(4) BEST
308.55/291.62	    (5) proven lower bound
308.55/291.62	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
308.55/291.62	        (7) BOUNDS(n^1, INF)
308.55/291.62	    (8) TRS for Loop Detection
308.55/291.62	
308.55/291.62	
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(0)
308.55/291.62	Obligation:
308.55/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.55/291.62	
308.55/291.62	
308.55/291.62	The TRS R consists of the following rules:
308.55/291.62	
308.55/291.62	   eq(0, 0) -> true
308.55/291.62	   eq(0, s(x)) -> false
308.55/291.62	   eq(s(x), 0) -> false
308.55/291.62	   eq(s(x), s(y)) -> eq(x, y)
308.55/291.62	   le(0, y) -> true
308.55/291.62	   le(s(x), 0) -> false
308.55/291.62	   le(s(x), s(y)) -> le(x, y)
308.55/291.62	   app(nil, y) -> y
308.55/291.62	   app(add(n, x), y) -> add(n, app(x, y))
308.55/291.62	   min(add(n, nil)) -> n
308.55/291.62	   min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
308.55/291.62	   if_min(true, add(n, add(m, x))) -> min(add(n, x))
308.55/291.62	   if_min(false, add(n, add(m, x))) -> min(add(m, x))
308.55/291.62	   head(add(n, x)) -> n
308.55/291.62	   tail(add(n, x)) -> x
308.55/291.62	   tail(nil) -> nil
308.55/291.62	   null(nil) -> true
308.55/291.62	   null(add(n, x)) -> false
308.55/291.62	   rm(n, nil) -> nil
308.55/291.62	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
308.55/291.62	   if_rm(true, n, add(m, x)) -> rm(n, x)
308.55/291.62	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
308.55/291.62	   minsort(x) -> mins(x, nil, nil)
308.55/291.62	   mins(x, y, z) -> if(null(x), x, y, z)
308.55/291.62	   if(true, x, y, z) -> z
308.55/291.62	   if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
308.55/291.62	   if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil)))
308.55/291.62	   if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z)
308.55/291.62	
308.55/291.62	S is empty.
308.55/291.62	Rewrite Strategy: FULL
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
308.55/291.62	Transformed a relative TRS into a decreasing-loop problem.
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(2)
308.55/291.62	Obligation:
308.55/291.62	Analyzing the following TRS for decreasing loops:
308.55/291.62	
308.55/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.55/291.62	
308.55/291.62	
308.55/291.62	The TRS R consists of the following rules:
308.55/291.62	
308.55/291.62	   eq(0, 0) -> true
308.55/291.62	   eq(0, s(x)) -> false
308.55/291.62	   eq(s(x), 0) -> false
308.55/291.62	   eq(s(x), s(y)) -> eq(x, y)
308.55/291.62	   le(0, y) -> true
308.55/291.62	   le(s(x), 0) -> false
308.55/291.62	   le(s(x), s(y)) -> le(x, y)
308.55/291.62	   app(nil, y) -> y
308.55/291.62	   app(add(n, x), y) -> add(n, app(x, y))
308.55/291.62	   min(add(n, nil)) -> n
308.55/291.62	   min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
308.55/291.62	   if_min(true, add(n, add(m, x))) -> min(add(n, x))
308.55/291.62	   if_min(false, add(n, add(m, x))) -> min(add(m, x))
308.55/291.62	   head(add(n, x)) -> n
308.55/291.62	   tail(add(n, x)) -> x
308.55/291.62	   tail(nil) -> nil
308.55/291.62	   null(nil) -> true
308.55/291.62	   null(add(n, x)) -> false
308.55/291.62	   rm(n, nil) -> nil
308.55/291.62	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
308.55/291.62	   if_rm(true, n, add(m, x)) -> rm(n, x)
308.55/291.62	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
308.55/291.62	   minsort(x) -> mins(x, nil, nil)
308.55/291.62	   mins(x, y, z) -> if(null(x), x, y, z)
308.55/291.62	   if(true, x, y, z) -> z
308.55/291.62	   if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
308.55/291.62	   if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil)))
308.55/291.62	   if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z)
308.55/291.62	
308.55/291.62	S is empty.
308.55/291.62	Rewrite Strategy: FULL
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(3) DecreasingLoopProof (LOWER BOUND(ID))
308.55/291.62	The following loop(s) give(s) rise to the lower bound Omega(n^1):
308.55/291.62	
308.55/291.62	The rewrite sequence
308.55/291.62	
308.55/291.62	le(s(x), s(y)) ->^+ le(x, y)
308.55/291.62	
308.55/291.62	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
308.55/291.62	
308.55/291.62	The pumping substitution is [x / s(x), y / s(y)].
308.55/291.62	
308.55/291.62	The result substitution is [ ].
308.55/291.62	
308.55/291.62	
308.55/291.62	
308.55/291.62	
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(4)
308.55/291.62	Complex Obligation (BEST)
308.55/291.62	
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(5)
308.55/291.62	Obligation:
308.55/291.62	Proved the lower bound n^1 for the following obligation:
308.55/291.62	
308.55/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.55/291.62	
308.55/291.62	
308.55/291.62	The TRS R consists of the following rules:
308.55/291.62	
308.55/291.62	   eq(0, 0) -> true
308.55/291.62	   eq(0, s(x)) -> false
308.55/291.62	   eq(s(x), 0) -> false
308.55/291.62	   eq(s(x), s(y)) -> eq(x, y)
308.55/291.62	   le(0, y) -> true
308.55/291.62	   le(s(x), 0) -> false
308.55/291.62	   le(s(x), s(y)) -> le(x, y)
308.55/291.62	   app(nil, y) -> y
308.55/291.62	   app(add(n, x), y) -> add(n, app(x, y))
308.55/291.62	   min(add(n, nil)) -> n
308.55/291.62	   min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
308.55/291.62	   if_min(true, add(n, add(m, x))) -> min(add(n, x))
308.55/291.62	   if_min(false, add(n, add(m, x))) -> min(add(m, x))
308.55/291.62	   head(add(n, x)) -> n
308.55/291.62	   tail(add(n, x)) -> x
308.55/291.62	   tail(nil) -> nil
308.55/291.62	   null(nil) -> true
308.55/291.62	   null(add(n, x)) -> false
308.55/291.62	   rm(n, nil) -> nil
308.55/291.62	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
308.55/291.62	   if_rm(true, n, add(m, x)) -> rm(n, x)
308.55/291.62	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
308.55/291.62	   minsort(x) -> mins(x, nil, nil)
308.55/291.62	   mins(x, y, z) -> if(null(x), x, y, z)
308.55/291.62	   if(true, x, y, z) -> z
308.55/291.62	   if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
308.55/291.62	   if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil)))
308.55/291.62	   if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z)
308.55/291.62	
308.55/291.62	S is empty.
308.55/291.62	Rewrite Strategy: FULL
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(6) LowerBoundPropagationProof (FINISHED)
308.55/291.62	Propagated lower bound.
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(7)
308.55/291.62	BOUNDS(n^1, INF)
308.55/291.62	
308.55/291.62	----------------------------------------
308.55/291.62	
308.55/291.62	(8)
308.55/291.62	Obligation:
308.55/291.62	Analyzing the following TRS for decreasing loops:
308.55/291.62	
308.55/291.62	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.55/291.62	
308.55/291.62	
308.55/291.62	The TRS R consists of the following rules:
308.55/291.62	
308.55/291.62	   eq(0, 0) -> true
308.55/291.62	   eq(0, s(x)) -> false
308.55/291.62	   eq(s(x), 0) -> false
308.55/291.62	   eq(s(x), s(y)) -> eq(x, y)
308.55/291.62	   le(0, y) -> true
308.55/291.62	   le(s(x), 0) -> false
308.55/291.62	   le(s(x), s(y)) -> le(x, y)
308.55/291.62	   app(nil, y) -> y
308.55/291.62	   app(add(n, x), y) -> add(n, app(x, y))
308.55/291.62	   min(add(n, nil)) -> n
308.55/291.62	   min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
308.55/291.62	   if_min(true, add(n, add(m, x))) -> min(add(n, x))
308.55/291.62	   if_min(false, add(n, add(m, x))) -> min(add(m, x))
308.55/291.62	   head(add(n, x)) -> n
308.55/291.62	   tail(add(n, x)) -> x
308.55/291.62	   tail(nil) -> nil
308.55/291.62	   null(nil) -> true
308.55/291.62	   null(add(n, x)) -> false
308.55/291.62	   rm(n, nil) -> nil
308.55/291.62	   rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
308.55/291.62	   if_rm(true, n, add(m, x)) -> rm(n, x)
308.55/291.62	   if_rm(false, n, add(m, x)) -> add(m, rm(n, x))
308.55/291.62	   minsort(x) -> mins(x, nil, nil)
308.55/291.62	   mins(x, y, z) -> if(null(x), x, y, z)
308.55/291.62	   if(true, x, y, z) -> z
308.55/291.62	   if(false, x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
308.55/291.62	   if2(true, x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil, app(z, add(head(x), nil)))
308.55/291.62	   if2(false, x, y, z) -> mins(tail(x), add(head(x), y), z)
308.55/291.62	
308.55/291.62	S is empty.
308.55/291.62	Rewrite Strategy: FULL
308.58/291.65	EOF