32.32/9.11	YES
32.32/9.12	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
32.32/9.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.32/9.12	
32.32/9.12	
32.32/9.12	Termination of the given RelTRS could be proven:
32.32/9.12	
32.32/9.12	(0) RelTRS
32.32/9.12	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
32.32/9.12	(2) RelTRS
32.32/9.12	(3) RelTRSRRRProof [EQUIVALENT, 143 ms]
32.32/9.12	(4) RelTRS
32.32/9.12	(5) RelTRSRRRProof [EQUIVALENT, 746 ms]
32.32/9.12	(6) RelTRS
32.32/9.12	(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
32.32/9.12	(8) RelTRS
32.32/9.12	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
32.32/9.12	(10) YES
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(0)
32.32/9.12	Obligation:
32.32/9.12	Relative term rewrite system:
32.32/9.12	The relative TRS consists of the following R rules:
32.32/9.12	
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	
32.32/9.12	The relative TRS consists of the following S rules:
32.32/9.12	
32.32/9.12	   0(p(0(x1))) -> q(x1)
32.32/9.12	   1(p(1(x1))) -> q(x1)
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(p(1(0(1(x1)))))
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(1) RelTRS Reverse (EQUIVALENT)
32.32/9.12	We have reversed the following relative TRS [REVERSE]:
32.32/9.12	The set of rules R is 
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	
32.32/9.12	The set of rules S is 
32.32/9.12	   0(p(0(x1))) -> q(x1)
32.32/9.12	   1(p(1(x1))) -> q(x1)
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(p(1(0(1(x1)))))
32.32/9.12	
32.32/9.12	We have obtained the following relative TRS:
32.32/9.12	The set of rules R is 
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	
32.32/9.12	The set of rules S is 
32.32/9.12	   0(p(0(x1))) -> q(x1)
32.32/9.12	   1(p(1(x1))) -> q(x1)
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(0(1(p(1(x1)))))
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(2)
32.32/9.12	Obligation:
32.32/9.12	Relative term rewrite system:
32.32/9.12	The relative TRS consists of the following R rules:
32.32/9.12	
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	
32.32/9.12	The relative TRS consists of the following S rules:
32.32/9.12	
32.32/9.12	   0(p(0(x1))) -> q(x1)
32.32/9.12	   1(p(1(x1))) -> q(x1)
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(0(1(p(1(x1)))))
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(3) RelTRSRRRProof (EQUIVALENT)
32.32/9.12	We used the following monotonic ordering for rule removal:
32.32/9.12	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(q(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(p(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(0(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(1(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
32.32/9.12	Rules from R:
32.32/9.12	none
32.32/9.12	Rules from S:
32.32/9.12	
32.32/9.12	   0(p(0(x1))) -> q(x1)
32.32/9.12	
32.32/9.12	
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(4)
32.32/9.12	Obligation:
32.32/9.12	Relative term rewrite system:
32.32/9.12	The relative TRS consists of the following R rules:
32.32/9.12	
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	
32.32/9.12	The relative TRS consists of the following S rules:
32.32/9.12	
32.32/9.12	   1(p(1(x1))) -> q(x1)
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(0(1(p(1(x1)))))
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(5) RelTRSRRRProof (EQUIVALENT)
32.32/9.12	We used the following monotonic ordering for rule removal:
32.32/9.12	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(b(x_1)) =  	[[0], [2], [0]] 	 +  	[[1, 0, 1], [2, 0, 2], [0, 0, 2]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(q(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 2, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(p(x_1)) =  	[[1], [0], [1]] 	 +  	[[1, 0, 0], [0, 0, 0], [2, 1, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(1(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 2, 0], [0, 0, 1]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	   <<<
32.32/9.12	 POL(0(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 2, 0], [0, 0, 0], [1, 0, 0]] 	* 	x_1
32.32/9.12	>>>
32.32/9.12	
32.32/9.12	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
32.32/9.12	Rules from R:
32.32/9.12	none
32.32/9.12	Rules from S:
32.32/9.12	
32.32/9.12	   1(p(1(x1))) -> q(x1)
32.32/9.12	
32.32/9.12	
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(6)
32.32/9.12	Obligation:
32.32/9.12	Relative term rewrite system:
32.32/9.12	The relative TRS consists of the following R rules:
32.32/9.12	
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	
32.32/9.12	The relative TRS consists of the following S rules:
32.32/9.12	
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(0(1(p(1(x1)))))
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(7) RelTRSRRRProof (EQUIVALENT)
32.32/9.12	We used the following monotonic ordering for rule removal:
32.32/9.12	Polynomial interpretation [POLO]:
32.32/9.12	
32.32/9.12	   POL(0(x_1)) = x_1
32.32/9.12	   POL(1(x_1)) = x_1
32.32/9.12	   POL(b(x_1)) = x_1
32.32/9.12	   POL(p(x_1)) = x_1
32.32/9.12	   POL(q(x_1)) = 1 + x_1
32.32/9.12	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
32.32/9.12	Rules from R:
32.32/9.12	
32.32/9.12	   b(q(b(x1))) -> b(p(b(x1)))
32.32/9.12	Rules from S:
32.32/9.12	none
32.32/9.12	
32.32/9.12	
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(8)
32.32/9.12	Obligation:
32.32/9.12	Relative term rewrite system:
32.32/9.12	R is empty.
32.32/9.12	The relative TRS consists of the following S rules:
32.32/9.12	
32.32/9.12	   0(q(0(x1))) -> q(x1)
32.32/9.12	   1(q(1(x1))) -> q(x1)
32.32/9.12	   p(x1) -> 1(0(1(p(1(x1)))))
32.32/9.12	
32.32/9.12	
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(9) RIsEmptyProof (EQUIVALENT)
32.32/9.12	The TRS R is empty. Hence, termination is trivially proven.
32.32/9.12	----------------------------------------
32.32/9.12	
32.32/9.12	(10)
32.32/9.12	YES
32.67/9.29	EOF