3.79/1.81	WORST_CASE(?, O(1))
3.79/1.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.79/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.79/1.84	
3.79/1.84	
3.79/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.79/1.84	
3.79/1.84	(0) CpxTRS
3.79/1.84	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
3.79/1.84	(2) CpxTRS
3.79/1.84	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.79/1.84	(4) BOUNDS(1, 1)
3.79/1.84	
3.79/1.84	
3.79/1.84	----------------------------------------
3.79/1.84	
3.79/1.84	(0)
3.79/1.84	Obligation:
3.79/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.79/1.84	
3.79/1.84	
3.79/1.84	The TRS R consists of the following rules:
3.79/1.84	
3.79/1.84	   f1 -> g1
3.79/1.84	   f1 -> g2
3.79/1.84	   f2 -> g1
3.79/1.84	   f2 -> g2
3.79/1.84	   g1 -> h1
3.79/1.84	   g1 -> h2
3.79/1.84	   g2 -> h1
3.79/1.84	   g2 -> h2
3.79/1.84	   h1 -> i
3.79/1.84	   h2 -> i
3.79/1.84	   e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
3.79/1.84	   e1(x1, x1, x, y, z) -> e5(x1, x, y, z)
3.79/1.84	   e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
3.79/1.84	   e2(x, x, y, z, z) -> e6(x, y, z)
3.79/1.84	   e2(i, x, y, z, i) -> e6(x, y, z)
3.79/1.84	   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
3.79/1.84	   e3(x, y, x, y, y, z, y, z, x, y, z) -> e6(x, y, z)
3.79/1.84	   e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)
3.79/1.84	   e4(i, x1, i, x1, i, x1, i, x1, x, y, z) -> e5(x1, x, y, z)
3.79/1.84	   e4(x, x, x, x, x, x, x, x, x, x, x) -> e6(x, x, x)
3.79/1.84	   e5(i, x, y, z) -> e6(x, y, z)
3.79/1.84	
3.79/1.84	S is empty.
3.79/1.84	Rewrite Strategy: FULL
3.79/1.84	----------------------------------------
3.79/1.84	
3.79/1.84	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
3.79/1.84	The TRS does not nest defined symbols.
3.79/1.84	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
3.79/1.84	e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
3.79/1.84	e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
3.79/1.84	e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)
3.79/1.84	
3.79/1.84	----------------------------------------
3.79/1.84	
3.79/1.84	(2)
3.79/1.84	Obligation:
3.79/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.79/1.84	
3.79/1.84	
3.79/1.84	The TRS R consists of the following rules:
3.79/1.84	
3.79/1.84	   f1 -> g1
3.79/1.84	   f1 -> g2
3.79/1.84	   f2 -> g1
3.79/1.84	   f2 -> g2
3.79/1.84	   g1 -> h1
3.79/1.84	   g1 -> h2
3.79/1.84	   g2 -> h1
3.79/1.84	   g2 -> h2
3.79/1.84	   h1 -> i
3.79/1.84	   h2 -> i
3.79/1.84	   e1(x1, x1, x, y, z) -> e5(x1, x, y, z)
3.79/1.84	   e2(x, x, y, z, z) -> e6(x, y, z)
3.79/1.84	   e2(i, x, y, z, i) -> e6(x, y, z)
3.79/1.84	   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
3.79/1.84	   e3(x, y, x, y, y, z, y, z, x, y, z) -> e6(x, y, z)
3.79/1.84	   e4(i, x1, i, x1, i, x1, i, x1, x, y, z) -> e5(x1, x, y, z)
3.79/1.84	   e4(x, x, x, x, x, x, x, x, x, x, x) -> e6(x, x, x)
3.79/1.84	   e5(i, x, y, z) -> e6(x, y, z)
3.79/1.84	
3.79/1.84	S is empty.
3.79/1.84	Rewrite Strategy: FULL
3.79/1.84	----------------------------------------
3.79/1.84	
3.79/1.84	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.79/1.84	Constant runtime complexity proven by termination of constructor-based narrowing.
3.79/1.84	
3.79/1.84	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.79/1.84	
3.79/1.84	h2 ->^* i
3.79/1.84	
3.79/1.84	g2 ->^* i
3.79/1.84	
3.79/1.84	e4(x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, x0) ->^* e6(x0, x0, x0)
3.79/1.84	
3.79/1.84	e4(i, i, i, i, i, i, i, i, x0, x1, x2) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	e3(x0, x1, x0, x1, x1, x2, x1, x2, x0, x1, x2) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	e3(x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, x0) ->^* e6(x0, x0, x0)
3.79/1.84	
3.79/1.84	e3(i, i, i, i, i, i, i, i, x0, x1, x2) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	e2(i, x0, x1, x2, i) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	e2(x0, x0, x1, x2, x2) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	e1(i, i, x0, x1, x2) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	e5(i, x0, x1, x2) ->^* e6(x0, x1, x2)
3.79/1.84	
3.79/1.84	h1 ->^* i
3.79/1.84	
3.79/1.84	f2 ->^* i
3.79/1.84	
3.79/1.84	g1 ->^* i
3.79/1.84	
3.79/1.84	f1 ->^* i
3.79/1.84	
3.79/1.84	
3.79/1.84	
3.79/1.84	
3.79/1.84	----------------------------------------
3.79/1.84	
3.79/1.84	(4)
3.79/1.84	BOUNDS(1, 1)
3.98/1.88	EOF