5.96/2.35	YES
5.96/2.36	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.96/2.36	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.96/2.36	
5.96/2.36	
5.96/2.36	Termination of the given RelTRS could be proven:
5.96/2.36	
5.96/2.36	(0) RelTRS
5.96/2.36	(1) RelTRSRRRProof [EQUIVALENT, 72 ms]
5.96/2.36	(2) RelTRS
5.96/2.36	(3) RelTRSRRRProof [EQUIVALENT, 10 ms]
5.96/2.36	(4) RelTRS
5.96/2.36	(5) RelTRSRRRProof [EQUIVALENT, 7 ms]
5.96/2.36	(6) RelTRS
5.96/2.36	(7) RIsEmptyProof [EQUIVALENT, 4 ms]
5.96/2.36	(8) YES
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(0)
5.96/2.36	Obligation:
5.96/2.36	Relative term rewrite system:
5.96/2.36	The relative TRS consists of the following R rules:
5.96/2.36	
5.96/2.36	   l(m(x)) -> m(l(x))
5.96/2.36	   m(r(x)) -> r(m(x))
5.96/2.36	   f(m(x), y) -> f(x, m(y))
5.96/2.36	
5.96/2.36	The relative TRS consists of the following S rules:
5.96/2.36	
5.96/2.36	   b -> l(b)
5.96/2.36	   f(x, y) -> f(x, r(y))
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(1) RelTRSRRRProof (EQUIVALENT)
5.96/2.36	We used the following monotonic ordering for rule removal:
5.96/2.36	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(l(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(m(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(r(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(f(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(b) =  	[[0], [0]]
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.96/2.36	Rules from R:
5.96/2.36	
5.96/2.36	   l(m(x)) -> m(l(x))
5.96/2.36	Rules from S:
5.96/2.36	none
5.96/2.36	
5.96/2.36	
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(2)
5.96/2.36	Obligation:
5.96/2.36	Relative term rewrite system:
5.96/2.36	The relative TRS consists of the following R rules:
5.96/2.36	
5.96/2.36	   m(r(x)) -> r(m(x))
5.96/2.36	   f(m(x), y) -> f(x, m(y))
5.96/2.36	
5.96/2.36	The relative TRS consists of the following S rules:
5.96/2.36	
5.96/2.36	   b -> l(b)
5.96/2.36	   f(x, y) -> f(x, r(y))
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(3) RelTRSRRRProof (EQUIVALENT)
5.96/2.36	We used the following monotonic ordering for rule removal:
5.96/2.36	m/1(YES)
5.96/2.36	r/1)YES(
5.96/2.36	f/2(YES,YES)
5.96/2.36	b/0)
5.96/2.36	l/1)YES(
5.96/2.36	
5.96/2.36	Quasi precedence:
5.96/2.36	f_2 > m_1
5.96/2.36	b > m_1
5.96/2.36	
5.96/2.36	
5.96/2.36	Status:
5.96/2.36	m_1: [1]
5.96/2.36	f_2: [1,2]
5.96/2.36	b: multiset status
5.96/2.36	
5.96/2.36	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.96/2.36	Rules from R:
5.96/2.36	
5.96/2.36	   f(m(x), y) -> f(x, m(y))
5.96/2.36	Rules from S:
5.96/2.36	none
5.96/2.36	
5.96/2.36	
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(4)
5.96/2.36	Obligation:
5.96/2.36	Relative term rewrite system:
5.96/2.36	The relative TRS consists of the following R rules:
5.96/2.36	
5.96/2.36	   m(r(x)) -> r(m(x))
5.96/2.36	
5.96/2.36	The relative TRS consists of the following S rules:
5.96/2.36	
5.96/2.36	   b -> l(b)
5.96/2.36	   f(x, y) -> f(x, r(y))
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(5) RelTRSRRRProof (EQUIVALENT)
5.96/2.36	We used the following monotonic ordering for rule removal:
5.96/2.36	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(m(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(r(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(b) =  	[[1], [1]]
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(l(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	   <<<
5.96/2.36	 POL(f(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2
5.96/2.36	>>>
5.96/2.36	
5.96/2.36	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.96/2.36	Rules from R:
5.96/2.36	
5.96/2.36	   m(r(x)) -> r(m(x))
5.96/2.36	Rules from S:
5.96/2.36	none
5.96/2.36	
5.96/2.36	
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(6)
5.96/2.36	Obligation:
5.96/2.36	Relative term rewrite system:
5.96/2.36	R is empty.
5.96/2.36	The relative TRS consists of the following S rules:
5.96/2.36	
5.96/2.36	   b -> l(b)
5.96/2.36	   f(x, y) -> f(x, r(y))
5.96/2.36	
5.96/2.36	
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(7) RIsEmptyProof (EQUIVALENT)
5.96/2.36	The TRS R is empty. Hence, termination is trivially proven.
5.96/2.36	----------------------------------------
5.96/2.36	
5.96/2.36	(8)
5.96/2.36	YES
6.26/2.41	EOF