309.24/291.51	WORST_CASE(Omega(n^1), ?)
309.24/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
309.24/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
309.24/291.52	
309.24/291.52	
309.24/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.24/291.52	
309.24/291.52	(0) CpxTRS
309.24/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
309.24/291.52	(2) TRS for Loop Detection
309.24/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
309.24/291.52	(4) BEST
309.24/291.52	    (5) proven lower bound
309.24/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
309.24/291.52	        (7) BOUNDS(n^1, INF)
309.24/291.52	    (8) TRS for Loop Detection
309.24/291.52	
309.24/291.52	
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(0)
309.24/291.52	Obligation:
309.24/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.24/291.52	
309.24/291.52	
309.24/291.52	The TRS R consists of the following rules:
309.24/291.52	
309.24/291.52	   eq(0, 0) -> true
309.24/291.52	   eq(0, s(X)) -> false
309.24/291.52	   eq(s(X), 0) -> false
309.24/291.52	   eq(s(X), s(Y)) -> eq(X, Y)
309.24/291.52	   rm(N, nil) -> nil
309.24/291.52	   rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
309.24/291.52	   ifrm(true, N, add(M, X)) -> rm(N, X)
309.24/291.52	   ifrm(false, N, add(M, X)) -> add(M, rm(N, X))
309.24/291.52	   purge(nil) -> nil
309.24/291.52	   purge(add(N, X)) -> add(N, purge(rm(N, X)))
309.24/291.52	
309.24/291.52	S is empty.
309.24/291.52	Rewrite Strategy: FULL
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
309.24/291.52	Transformed a relative TRS into a decreasing-loop problem.
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(2)
309.24/291.52	Obligation:
309.24/291.52	Analyzing the following TRS for decreasing loops:
309.24/291.52	
309.24/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.24/291.52	
309.24/291.52	
309.24/291.52	The TRS R consists of the following rules:
309.24/291.52	
309.24/291.52	   eq(0, 0) -> true
309.24/291.52	   eq(0, s(X)) -> false
309.24/291.52	   eq(s(X), 0) -> false
309.24/291.52	   eq(s(X), s(Y)) -> eq(X, Y)
309.24/291.52	   rm(N, nil) -> nil
309.24/291.52	   rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
309.24/291.52	   ifrm(true, N, add(M, X)) -> rm(N, X)
309.24/291.52	   ifrm(false, N, add(M, X)) -> add(M, rm(N, X))
309.24/291.52	   purge(nil) -> nil
309.24/291.52	   purge(add(N, X)) -> add(N, purge(rm(N, X)))
309.24/291.52	
309.24/291.52	S is empty.
309.24/291.52	Rewrite Strategy: FULL
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
309.24/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
309.24/291.52	
309.24/291.52	The rewrite sequence
309.24/291.52	
309.24/291.52	eq(s(X), s(Y)) ->^+ eq(X, Y)
309.24/291.52	
309.24/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
309.24/291.52	
309.24/291.52	The pumping substitution is [X / s(X), Y / s(Y)].
309.24/291.52	
309.24/291.52	The result substitution is [ ].
309.24/291.52	
309.24/291.52	
309.24/291.52	
309.24/291.52	
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(4)
309.24/291.52	Complex Obligation (BEST)
309.24/291.52	
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(5)
309.24/291.52	Obligation:
309.24/291.52	Proved the lower bound n^1 for the following obligation:
309.24/291.52	
309.24/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.24/291.52	
309.24/291.52	
309.24/291.52	The TRS R consists of the following rules:
309.24/291.52	
309.24/291.52	   eq(0, 0) -> true
309.24/291.52	   eq(0, s(X)) -> false
309.24/291.52	   eq(s(X), 0) -> false
309.24/291.52	   eq(s(X), s(Y)) -> eq(X, Y)
309.24/291.52	   rm(N, nil) -> nil
309.24/291.52	   rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
309.24/291.52	   ifrm(true, N, add(M, X)) -> rm(N, X)
309.24/291.52	   ifrm(false, N, add(M, X)) -> add(M, rm(N, X))
309.24/291.52	   purge(nil) -> nil
309.24/291.52	   purge(add(N, X)) -> add(N, purge(rm(N, X)))
309.24/291.52	
309.24/291.52	S is empty.
309.24/291.52	Rewrite Strategy: FULL
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(6) LowerBoundPropagationProof (FINISHED)
309.24/291.52	Propagated lower bound.
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(7)
309.24/291.52	BOUNDS(n^1, INF)
309.24/291.52	
309.24/291.52	----------------------------------------
309.24/291.52	
309.24/291.52	(8)
309.24/291.52	Obligation:
309.24/291.52	Analyzing the following TRS for decreasing loops:
309.24/291.52	
309.24/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.24/291.52	
309.24/291.52	
309.24/291.52	The TRS R consists of the following rules:
309.24/291.52	
309.24/291.52	   eq(0, 0) -> true
309.24/291.52	   eq(0, s(X)) -> false
309.24/291.52	   eq(s(X), 0) -> false
309.24/291.52	   eq(s(X), s(Y)) -> eq(X, Y)
309.24/291.52	   rm(N, nil) -> nil
309.24/291.52	   rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
309.24/291.52	   ifrm(true, N, add(M, X)) -> rm(N, X)
309.24/291.52	   ifrm(false, N, add(M, X)) -> add(M, rm(N, X))
309.24/291.52	   purge(nil) -> nil
309.24/291.52	   purge(add(N, X)) -> add(N, purge(rm(N, X)))
309.24/291.52	
309.24/291.52	S is empty.
309.24/291.52	Rewrite Strategy: FULL
309.33/291.55	EOF