3.72/1.70	WORST_CASE(?, O(1))
3.72/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.72/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.72/1.71	
3.72/1.71	
3.72/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.72/1.71	
3.72/1.71	(0) CpxTRS
3.72/1.71	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
3.72/1.71	(2) CpxTRS
3.72/1.71	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.72/1.71	(4) BOUNDS(1, 1)
3.72/1.71	
3.72/1.71	
3.72/1.71	----------------------------------------
3.72/1.71	
3.72/1.71	(0)
3.72/1.71	Obligation:
3.72/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.72/1.71	
3.72/1.71	
3.72/1.71	The TRS R consists of the following rules:
3.72/1.71	
3.72/1.71	   f1(a, x) -> g1(x, x)
3.72/1.71	   f1(x, a) -> g2(x, x)
3.72/1.71	   f2(a, x) -> g1(x, x)
3.72/1.71	   f2(x, a) -> g2(x, x)
3.72/1.71	   g1(a, x) -> h1(x)
3.72/1.71	   g1(x, a) -> h2(x)
3.72/1.71	   g2(a, x) -> h1(x)
3.72/1.71	   g2(x, a) -> h2(x)
3.72/1.71	   h1(a) -> i
3.72/1.71	   h2(a) -> i
3.72/1.71	   e1(h1(w), h2(w), x, y, z, w) -> e2(x, x, y, z, z, w)
3.72/1.71	   e1(x1, x1, x, y, z, a) -> e5(x1, x, y, z)
3.72/1.71	   e2(f1(w, w), x, y, z, f2(w, w), w) -> e3(x, y, x, y, y, z, y, z, x, y, z, w)
3.72/1.71	   e2(x, x, y, z, z, a) -> e6(x, y, z)
3.72/1.71	   e2(i, x, y, z, i, a) -> e6(x, y, z)
3.72/1.71	   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w)
3.72/1.71	   e3(x, y, x, y, y, z, y, z, x, y, z, a) -> e6(x, y, z)
3.72/1.71	   e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) -> e1(x1, x1, x, y, z, w)
3.72/1.71	   e4(i, x1, i, x1, i, x1, i, x1, x, y, z, a) -> e5(x1, x, y, z)
3.72/1.71	   e4(x, x, x, x, x, x, x, x, x, x, x, a) -> e6(x, x, x)
3.72/1.71	   e5(i, x, y, z) -> e6(x, y, z)
3.72/1.71	
3.72/1.71	S is empty.
3.72/1.71	Rewrite Strategy: FULL
3.72/1.71	----------------------------------------
3.72/1.71	
3.72/1.71	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
3.72/1.71	The TRS does not nest defined symbols.
3.72/1.71	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
3.72/1.71	e1(h1(w), h2(w), x, y, z, w) -> e2(x, x, y, z, z, w)
3.72/1.71	e2(f1(w, w), x, y, z, f2(w, w), w) -> e3(x, y, x, y, y, z, y, z, x, y, z, w)
3.72/1.71	e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) -> e1(x1, x1, x, y, z, w)
3.72/1.71	
3.72/1.71	----------------------------------------
3.72/1.71	
3.72/1.71	(2)
3.72/1.71	Obligation:
3.72/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.72/1.71	
3.72/1.71	
3.72/1.71	The TRS R consists of the following rules:
3.72/1.71	
3.72/1.71	   f1(a, x) -> g1(x, x)
3.72/1.71	   f1(x, a) -> g2(x, x)
3.72/1.71	   f2(a, x) -> g1(x, x)
3.72/1.71	   f2(x, a) -> g2(x, x)
3.72/1.71	   g1(a, x) -> h1(x)
3.72/1.71	   g1(x, a) -> h2(x)
3.72/1.71	   g2(a, x) -> h1(x)
3.72/1.71	   g2(x, a) -> h2(x)
3.72/1.71	   h1(a) -> i
3.72/1.71	   h2(a) -> i
3.72/1.71	   e1(x1, x1, x, y, z, a) -> e5(x1, x, y, z)
3.72/1.71	   e2(x, x, y, z, z, a) -> e6(x, y, z)
3.72/1.71	   e2(i, x, y, z, i, a) -> e6(x, y, z)
3.72/1.71	   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w)
3.72/1.71	   e3(x, y, x, y, y, z, y, z, x, y, z, a) -> e6(x, y, z)
3.72/1.71	   e4(i, x1, i, x1, i, x1, i, x1, x, y, z, a) -> e5(x1, x, y, z)
3.72/1.71	   e4(x, x, x, x, x, x, x, x, x, x, x, a) -> e6(x, x, x)
3.72/1.71	   e5(i, x, y, z) -> e6(x, y, z)
3.72/1.71	
3.72/1.71	S is empty.
3.72/1.71	Rewrite Strategy: FULL
3.72/1.71	----------------------------------------
3.72/1.71	
3.72/1.71	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.72/1.71	Constant runtime complexity proven by termination of constructor-based narrowing.
3.72/1.71	
3.72/1.71	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.72/1.71	
3.72/1.71	h2(a) ->^* i
3.72/1.71	
3.72/1.71	g2(a, a) ->^* i
3.72/1.71	
3.72/1.71	e4(x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, a) ->^* e6(x0, x0, x0)
3.72/1.71	
3.72/1.71	e4(i, i, i, i, i, i, i, i, x0, x1, x2, a) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	e3(x0, x1, x0, x1, x1, x2, x1, x2, x0, x1, x2, a) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	e3(x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, a) ->^* e6(x0, x0, x0)
3.72/1.71	
3.72/1.71	e3(i, i, i, i, i, i, i, i, x0, x1, x2, a) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	e2(i, x0, x1, x2, i, a) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	e2(x0, x0, x1, x2, x2, a) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	e1(i, i, x0, x1, x2, a) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	e5(i, x0, x1, x2) ->^* e6(x0, x1, x2)
3.72/1.71	
3.72/1.71	h1(a) ->^* i
3.72/1.71	
3.72/1.71	f2(a, a) ->^* i
3.72/1.71	
3.72/1.71	g1(a, a) ->^* i
3.72/1.71	
3.72/1.71	f1(a, a) ->^* i
3.72/1.71	
3.72/1.71	
3.72/1.71	
3.72/1.71	
3.72/1.71	----------------------------------------
3.72/1.71	
3.72/1.71	(4)
3.72/1.71	BOUNDS(1, 1)
3.89/1.78	EOF