53.23/14.31	YES
53.23/14.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
53.23/14.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
53.23/14.32	
53.23/14.32	
53.23/14.32	Termination of the given RelTRS could be proven:
53.23/14.32	
53.23/14.32	(0) RelTRS
53.23/14.32	(1) RelTRSRRRProof [EQUIVALENT, 2054 ms]
53.23/14.32	(2) RelTRS
53.23/14.32	(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
53.23/14.32	(4) RelTRS
53.23/14.32	(5) RelTRSRRRProof [EQUIVALENT, 3 ms]
53.23/14.32	(6) RelTRS
53.23/14.32	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
53.23/14.32	(8) YES
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(0)
53.23/14.32	Obligation:
53.23/14.32	Relative term rewrite system:
53.23/14.32	The relative TRS consists of the following R rules:
53.23/14.32	
53.23/14.32	   c(c(a(x1))) -> b(b(c(x1)))
53.23/14.32	   a(b(c(x1))) -> a(b(a(x1)))
53.23/14.32	
53.23/14.32	The relative TRS consists of the following S rules:
53.23/14.32	
53.23/14.32	   c(c(c(x1))) -> b(c(a(x1)))
53.23/14.32	   a(b(b(x1))) -> a(c(a(x1)))
53.23/14.32	   c(c(a(x1))) -> b(b(b(x1)))
53.23/14.32	   c(b(a(x1))) -> c(c(b(x1)))
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(1) RelTRSRRRProof (EQUIVALENT)
53.23/14.32	We used the following monotonic ordering for rule removal:
53.23/14.32	Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
53.23/14.32	
53.23/14.32	   <<<
53.23/14.32	 POL(c(x_1)) =  	[[0], [1], [0], [1], [0], [0]] 	 +  	[[1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
53.23/14.32	>>>
53.23/14.32	
53.23/14.32	   <<<
53.23/14.32	 POL(a(x_1)) =  	[[0], [0], [0], [1], [1], [0]] 	 +  	[[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
53.23/14.32	>>>
53.23/14.32	
53.23/14.32	   <<<
53.23/14.32	 POL(b(x_1)) =  	[[0], [1], [0], [1], [0], [0]] 	 +  	[[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
53.23/14.32	>>>
53.23/14.32	
53.23/14.32	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
53.23/14.32	Rules from R:
53.23/14.32	
53.23/14.32	   c(c(a(x1))) -> b(b(c(x1)))
53.23/14.32	Rules from S:
53.23/14.32	
53.23/14.32	   c(c(c(x1))) -> b(c(a(x1)))
53.23/14.32	   c(c(a(x1))) -> b(b(b(x1)))
53.23/14.32	
53.23/14.32	
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(2)
53.23/14.32	Obligation:
53.23/14.32	Relative term rewrite system:
53.23/14.32	The relative TRS consists of the following R rules:
53.23/14.32	
53.23/14.32	   a(b(c(x1))) -> a(b(a(x1)))
53.23/14.32	
53.23/14.32	The relative TRS consists of the following S rules:
53.23/14.32	
53.23/14.32	   a(b(b(x1))) -> a(c(a(x1)))
53.23/14.32	   c(b(a(x1))) -> c(c(b(x1)))
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(3) RelTRSRRRProof (EQUIVALENT)
53.23/14.32	We used the following monotonic ordering for rule removal:
53.23/14.32	Polynomial interpretation [POLO]:
53.23/14.32	
53.23/14.32	   POL(a(x_1)) = x_1
53.23/14.32	   POL(b(x_1)) = 1 + x_1
53.23/14.32	   POL(c(x_1)) = x_1
53.23/14.32	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
53.23/14.32	Rules from R:
53.23/14.32	none
53.23/14.32	Rules from S:
53.23/14.32	
53.23/14.32	   a(b(b(x1))) -> a(c(a(x1)))
53.23/14.32	
53.23/14.32	
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(4)
53.23/14.32	Obligation:
53.23/14.32	Relative term rewrite system:
53.23/14.32	The relative TRS consists of the following R rules:
53.23/14.32	
53.23/14.32	   a(b(c(x1))) -> a(b(a(x1)))
53.23/14.32	
53.23/14.32	The relative TRS consists of the following S rules:
53.23/14.32	
53.23/14.32	   c(b(a(x1))) -> c(c(b(x1)))
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(5) RelTRSRRRProof (EQUIVALENT)
53.23/14.32	We used the following monotonic ordering for rule removal:
53.23/14.32	Knuth-Bendix order [KBO] with precedence:b_1 > c_1 > a_1
53.23/14.32	
53.23/14.32	and weight map:
53.23/14.32	
53.23/14.32	   a_1=2
53.23/14.32	   b_1=1
53.23/14.32	   c_1=2
53.23/14.32	
53.23/14.32	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
53.23/14.32	Rules from R:
53.23/14.32	
53.23/14.32	   a(b(c(x1))) -> a(b(a(x1)))
53.23/14.32	Rules from S:
53.23/14.32	
53.23/14.32	   c(b(a(x1))) -> c(c(b(x1)))
53.23/14.32	
53.23/14.32	
53.23/14.32	
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(6)
53.23/14.32	Obligation:
53.23/14.32	Relative term rewrite system:
53.23/14.32	R is empty.
53.23/14.32	S is empty.
53.23/14.32	
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(7) RIsEmptyProof (EQUIVALENT)
53.23/14.32	The TRS R is empty. Hence, termination is trivially proven.
53.23/14.32	----------------------------------------
53.23/14.32	
53.23/14.32	(8)
53.23/14.32	YES
53.44/14.35	EOF