12.61/4.05	YES
12.67/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.67/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.67/4.06	
12.67/4.06	
12.67/4.06	Termination of the given RelTRS could be proven:
12.67/4.06	
12.67/4.06	(0) RelTRS
12.67/4.06	(1) RelTRSRRRProof [EQUIVALENT, 115 ms]
12.67/4.06	(2) RelTRS
12.67/4.06	(3) RelTRSRRRProof [EQUIVALENT, 16 ms]
12.67/4.06	(4) RelTRS
12.67/4.06	(5) RelTRSRRRProof [EQUIVALENT, 4 ms]
12.67/4.06	(6) RelTRS
12.67/4.06	(7) RIsEmptyProof [EQUIVALENT, 1 ms]
12.67/4.06	(8) YES
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(0)
12.67/4.06	Obligation:
12.67/4.06	Relative term rewrite system:
12.67/4.06	The relative TRS consists of the following R rules:
12.67/4.06	
12.67/4.06	   b(c(a(x1))) -> d(d(x1))
12.67/4.06	   b(x1) -> c(c(x1))
12.67/4.06	   a(a(x1)) -> a(x1)
12.67/4.06	
12.67/4.06	The relative TRS consists of the following S rules:
12.67/4.06	
12.67/4.06	   a(b(x1)) -> d(x1)
12.67/4.06	   d(x1) -> a(b(x1))
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(1) RelTRSRRRProof (EQUIVALENT)
12.67/4.06	We used the following monotonic ordering for rule removal:
12.67/4.06	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(b(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(c(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 1], [0, 0, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(a(x_1)) =  	[[0], [0], [2]] 	 +  	[[1, 2, 2], [0, 0, 0], [0, 0, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(d(x_1)) =  	[[0], [0], [2]] 	 +  	[[1, 2, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
12.67/4.06	Rules from R:
12.67/4.06	
12.67/4.06	   a(a(x1)) -> a(x1)
12.67/4.06	Rules from S:
12.67/4.06	none
12.67/4.06	
12.67/4.06	
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(2)
12.67/4.06	Obligation:
12.67/4.06	Relative term rewrite system:
12.67/4.06	The relative TRS consists of the following R rules:
12.67/4.06	
12.67/4.06	   b(c(a(x1))) -> d(d(x1))
12.67/4.06	   b(x1) -> c(c(x1))
12.67/4.06	
12.67/4.06	The relative TRS consists of the following S rules:
12.67/4.06	
12.67/4.06	   a(b(x1)) -> d(x1)
12.67/4.06	   d(x1) -> a(b(x1))
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(3) RelTRSRRRProof (EQUIVALENT)
12.67/4.06	We used the following monotonic ordering for rule removal:
12.67/4.06	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(b(x_1)) =  	[[0], [2], [0]] 	 +  	[[1, 2, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(c(x_1)) =  	[[0], [2], [0]] 	 +  	[[1, 0, 0], [0, 0, 2], [0, 0, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(a(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	   <<<
12.67/4.06	 POL(d(x_1)) =  	[[0], [0], [3]] 	 +  	[[1, 2, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
12.67/4.06	>>>
12.67/4.06	
12.67/4.06	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
12.67/4.06	Rules from R:
12.67/4.06	
12.67/4.06	   b(c(a(x1))) -> d(d(x1))
12.67/4.06	Rules from S:
12.67/4.06	none
12.67/4.06	
12.67/4.06	
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(4)
12.67/4.06	Obligation:
12.67/4.06	Relative term rewrite system:
12.67/4.06	The relative TRS consists of the following R rules:
12.67/4.06	
12.67/4.06	   b(x1) -> c(c(x1))
12.67/4.06	
12.67/4.06	The relative TRS consists of the following S rules:
12.67/4.06	
12.67/4.06	   a(b(x1)) -> d(x1)
12.67/4.06	   d(x1) -> a(b(x1))
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(5) RelTRSRRRProof (EQUIVALENT)
12.67/4.06	We used the following monotonic ordering for rule removal:
12.67/4.06	Polynomial interpretation [POLO]:
12.67/4.06	
12.67/4.06	   POL(a(x_1)) = x_1
12.67/4.06	   POL(b(x_1)) = 1 + x_1
12.67/4.06	   POL(c(x_1)) = x_1
12.67/4.06	   POL(d(x_1)) = 1 + x_1
12.67/4.06	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
12.67/4.06	Rules from R:
12.67/4.06	
12.67/4.06	   b(x1) -> c(c(x1))
12.67/4.06	Rules from S:
12.67/4.06	none
12.67/4.06	
12.67/4.06	
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(6)
12.67/4.06	Obligation:
12.67/4.06	Relative term rewrite system:
12.67/4.06	R is empty.
12.67/4.06	The relative TRS consists of the following S rules:
12.67/4.06	
12.67/4.06	   a(b(x1)) -> d(x1)
12.67/4.06	   d(x1) -> a(b(x1))
12.67/4.06	
12.67/4.06	
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(7) RIsEmptyProof (EQUIVALENT)
12.67/4.06	The TRS R is empty. Hence, termination is trivially proven.
12.67/4.06	----------------------------------------
12.67/4.06	
12.67/4.06	(8)
12.67/4.06	YES
12.72/4.13	EOF