12.03/3.95	YES
16.63/5.15	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
16.63/5.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.63/5.15	
16.63/5.15	
16.63/5.15	Termination of the given RelTRS could be proven:
16.63/5.15	
16.63/5.15	(0) RelTRS
16.63/5.15	(1) RelTRSRRRProof [EQUIVALENT, 50 ms]
16.63/5.15	(2) RelTRS
16.63/5.15	(3) RelTRSRRRProof [EQUIVALENT, 52 ms]
16.63/5.15	(4) RelTRS
16.63/5.15	(5) RIsEmptyProof [EQUIVALENT, 1 ms]
16.63/5.15	(6) YES
16.63/5.15	
16.63/5.15	
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(0)
16.63/5.15	Obligation:
16.63/5.15	Relative term rewrite system:
16.63/5.15	The relative TRS consists of the following R rules:
16.63/5.15	
16.63/5.15	   top(ok(new(x))) -> top(check(x))
16.63/5.15	   top(ok(old(x))) -> top(check(x))
16.63/5.15	
16.63/5.15	The relative TRS consists of the following S rules:
16.63/5.15	
16.63/5.15	   bot -> new(bot)
16.63/5.15	   check(new(x)) -> new(check(x))
16.63/5.15	   check(old(x)) -> old(check(x))
16.63/5.15	   check(old(x)) -> ok(old(x))
16.63/5.15	   new(ok(x)) -> ok(new(x))
16.63/5.15	   old(ok(x)) -> ok(old(x))
16.63/5.15	
16.63/5.15	
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(1) RelTRSRRRProof (EQUIVALENT)
16.63/5.15	We used the following monotonic ordering for rule removal:
16.63/5.15	Polynomial interpretation [POLO]:
16.63/5.15	
16.63/5.15	   POL(bot) = 0
16.63/5.15	   POL(check(x_1)) = x_1
16.63/5.15	   POL(new(x_1)) = x_1
16.63/5.15	   POL(ok(x_1)) = x_1
16.63/5.15	   POL(old(x_1)) = 1 + x_1
16.63/5.15	   POL(top(x_1)) = x_1
16.63/5.15	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
16.63/5.15	Rules from R:
16.63/5.15	
16.63/5.15	   top(ok(old(x))) -> top(check(x))
16.63/5.15	Rules from S:
16.63/5.15	none
16.63/5.15	
16.63/5.15	
16.63/5.15	
16.63/5.15	
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(2)
16.63/5.15	Obligation:
16.63/5.15	Relative term rewrite system:
16.63/5.15	The relative TRS consists of the following R rules:
16.63/5.15	
16.63/5.15	   top(ok(new(x))) -> top(check(x))
16.63/5.15	
16.63/5.15	The relative TRS consists of the following S rules:
16.63/5.15	
16.63/5.15	   bot -> new(bot)
16.63/5.15	   check(new(x)) -> new(check(x))
16.63/5.15	   check(old(x)) -> old(check(x))
16.63/5.15	   check(old(x)) -> ok(old(x))
16.63/5.15	   new(ok(x)) -> ok(new(x))
16.63/5.15	   old(ok(x)) -> ok(old(x))
16.63/5.15	
16.63/5.15	
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(3) RelTRSRRRProof (EQUIVALENT)
16.63/5.15	We used the following monotonic ordering for rule removal:
16.63/5.15	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
16.63/5.15	
16.63/5.15	   <<<
16.63/5.15	 POL(top(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
16.63/5.15	>>>
16.63/5.15	
16.63/5.15	   <<<
16.63/5.15	 POL(ok(x_1)) =  	[[2], [2]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
16.63/5.15	>>>
16.63/5.15	
16.63/5.15	   <<<
16.63/5.15	 POL(new(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
16.63/5.15	>>>
16.63/5.15	
16.63/5.15	   <<<
16.63/5.15	 POL(check(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
16.63/5.15	>>>
16.63/5.15	
16.63/5.15	   <<<
16.63/5.15	 POL(bot) =  	[[0], [0]]
16.63/5.15	>>>
16.63/5.15	
16.63/5.15	   <<<
16.63/5.15	 POL(old(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
16.63/5.15	>>>
16.63/5.15	
16.63/5.15	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
16.63/5.15	Rules from R:
16.63/5.15	
16.63/5.15	   top(ok(new(x))) -> top(check(x))
16.63/5.15	Rules from S:
16.63/5.15	none
16.63/5.15	
16.63/5.15	
16.63/5.15	
16.63/5.15	
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(4)
16.63/5.15	Obligation:
16.63/5.15	Relative term rewrite system:
16.63/5.15	R is empty.
16.63/5.15	The relative TRS consists of the following S rules:
16.63/5.15	
16.63/5.15	   bot -> new(bot)
16.63/5.15	   check(new(x)) -> new(check(x))
16.63/5.15	   check(old(x)) -> old(check(x))
16.63/5.15	   check(old(x)) -> ok(old(x))
16.63/5.15	   new(ok(x)) -> ok(new(x))
16.63/5.15	   old(ok(x)) -> ok(old(x))
16.63/5.15	
16.63/5.15	
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(5) RIsEmptyProof (EQUIVALENT)
16.63/5.15	The TRS R is empty. Hence, termination is trivially proven.
16.63/5.15	----------------------------------------
16.63/5.15	
16.63/5.15	(6)
16.63/5.15	YES
16.63/5.20	EOF