8.00/3.00	YES
8.35/3.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.35/3.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.35/3.03	
8.35/3.03	
8.35/3.03	Termination of the given RelTRS could be proven:
8.35/3.03	
8.35/3.03	(0) RelTRS
8.35/3.03	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
8.35/3.03	(2) RelTRS
8.35/3.03	(3) RelTRSRRRProof [EQUIVALENT, 41 ms]
8.35/3.03	(4) RelTRS
8.35/3.03	(5) RelTRSRRRProof [EQUIVALENT, 14 ms]
8.35/3.03	(6) RelTRS
8.35/3.03	(7) RelTRSRRRProof [EQUIVALENT, 160 ms]
8.35/3.03	(8) RelTRS
8.35/3.03	(9) RIsEmptyProof [EQUIVALENT, 2 ms]
8.35/3.03	(10) YES
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(0)
8.35/3.03	Obligation:
8.35/3.03	Relative term rewrite system:
8.35/3.03	The relative TRS consists of the following R rules:
8.35/3.03	
8.35/3.03	   n(s(x1)) -> s(x1)
8.35/3.03	   o(s(x1)) -> s(x1)
8.35/3.03	   n(o(p(x1))) -> o(n(x1))
8.35/3.03	
8.35/3.03	The relative TRS consists of the following S rules:
8.35/3.03	
8.35/3.03	   t(x1) -> t(c(n(x1)))
8.35/3.03	   p(s(x1)) -> s(x1)
8.35/3.03	   o(n(x1)) -> n(o(x1))
8.35/3.03	   p(n(x1)) -> m(p(x1))
8.35/3.03	   p(m(x1)) -> m(p(x1))
8.35/3.03	   o(m(x1)) -> n(o(x1))
8.35/3.03	   n(p(x1)) -> p(n(x1))
8.35/3.03	   c(p(x1)) -> p(c(x1))
8.35/3.03	   c(m(x1)) -> m(c(x1))
8.35/3.03	   c(n(x1)) -> n(c(x1))
8.35/3.03	   c(o(x1)) -> o(c(x1))
8.35/3.03	   c(o(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(1) RelTRS Reverse (EQUIVALENT)
8.35/3.03	We have reversed the following relative TRS [REVERSE]:
8.35/3.03	The set of rules R is 
8.35/3.03	   n(s(x1)) -> s(x1)
8.35/3.03	   o(s(x1)) -> s(x1)
8.35/3.03	   n(o(p(x1))) -> o(n(x1))
8.35/3.03	
8.35/3.03	The set of rules S is 
8.35/3.03	   t(x1) -> t(c(n(x1)))
8.35/3.03	   p(s(x1)) -> s(x1)
8.35/3.03	   o(n(x1)) -> n(o(x1))
8.35/3.03	   p(n(x1)) -> m(p(x1))
8.35/3.03	   p(m(x1)) -> m(p(x1))
8.35/3.03	   o(m(x1)) -> n(o(x1))
8.35/3.03	   n(p(x1)) -> p(n(x1))
8.35/3.03	   c(p(x1)) -> p(c(x1))
8.35/3.03	   c(m(x1)) -> m(c(x1))
8.35/3.03	   c(n(x1)) -> n(c(x1))
8.35/3.03	   c(o(x1)) -> o(c(x1))
8.35/3.03	   c(o(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	We have obtained the following relative TRS:
8.35/3.03	The set of rules R is 
8.35/3.03	   s(n(x1)) -> s(x1)
8.35/3.03	   s(o(x1)) -> s(x1)
8.35/3.03	   p(o(n(x1))) -> n(o(x1))
8.35/3.03	
8.35/3.03	The set of rules S is 
8.35/3.03	   t(x1) -> n(c(t(x1)))
8.35/3.03	   s(p(x1)) -> s(x1)
8.35/3.03	   n(o(x1)) -> o(n(x1))
8.35/3.03	   n(p(x1)) -> p(m(x1))
8.35/3.03	   m(p(x1)) -> p(m(x1))
8.35/3.03	   m(o(x1)) -> o(n(x1))
8.35/3.03	   p(n(x1)) -> n(p(x1))
8.35/3.03	   p(c(x1)) -> c(p(x1))
8.35/3.03	   m(c(x1)) -> c(m(x1))
8.35/3.03	   n(c(x1)) -> c(n(x1))
8.35/3.03	   o(c(x1)) -> c(o(x1))
8.35/3.03	   o(c(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(2)
8.35/3.03	Obligation:
8.35/3.03	Relative term rewrite system:
8.35/3.03	The relative TRS consists of the following R rules:
8.35/3.03	
8.35/3.03	   s(n(x1)) -> s(x1)
8.35/3.03	   s(o(x1)) -> s(x1)
8.35/3.03	   p(o(n(x1))) -> n(o(x1))
8.35/3.03	
8.35/3.03	The relative TRS consists of the following S rules:
8.35/3.03	
8.35/3.03	   t(x1) -> n(c(t(x1)))
8.35/3.03	   s(p(x1)) -> s(x1)
8.35/3.03	   n(o(x1)) -> o(n(x1))
8.35/3.03	   n(p(x1)) -> p(m(x1))
8.35/3.03	   m(p(x1)) -> p(m(x1))
8.35/3.03	   m(o(x1)) -> o(n(x1))
8.35/3.03	   p(n(x1)) -> n(p(x1))
8.35/3.03	   p(c(x1)) -> c(p(x1))
8.35/3.03	   m(c(x1)) -> c(m(x1))
8.35/3.03	   n(c(x1)) -> c(n(x1))
8.35/3.03	   o(c(x1)) -> c(o(x1))
8.35/3.03	   o(c(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(3) RelTRSRRRProof (EQUIVALENT)
8.35/3.03	We used the following monotonic ordering for rule removal:
8.35/3.03	Polynomial interpretation [POLO]:
8.35/3.03	
8.35/3.03	   POL(c(x_1)) = x_1
8.35/3.03	   POL(m(x_1)) = x_1
8.35/3.03	   POL(n(x_1)) = x_1
8.35/3.03	   POL(o(x_1)) = x_1
8.35/3.03	   POL(p(x_1)) = 1 + x_1
8.35/3.03	   POL(s(x_1)) = x_1
8.35/3.03	   POL(t(x_1)) = x_1
8.35/3.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.35/3.03	Rules from R:
8.35/3.03	
8.35/3.03	   p(o(n(x1))) -> n(o(x1))
8.35/3.03	Rules from S:
8.35/3.03	
8.35/3.03	   s(p(x1)) -> s(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(4)
8.35/3.03	Obligation:
8.35/3.03	Relative term rewrite system:
8.35/3.03	The relative TRS consists of the following R rules:
8.35/3.03	
8.35/3.03	   s(n(x1)) -> s(x1)
8.35/3.03	   s(o(x1)) -> s(x1)
8.35/3.03	
8.35/3.03	The relative TRS consists of the following S rules:
8.35/3.03	
8.35/3.03	   t(x1) -> n(c(t(x1)))
8.35/3.03	   n(o(x1)) -> o(n(x1))
8.35/3.03	   n(p(x1)) -> p(m(x1))
8.35/3.03	   m(p(x1)) -> p(m(x1))
8.35/3.03	   m(o(x1)) -> o(n(x1))
8.35/3.03	   p(n(x1)) -> n(p(x1))
8.35/3.03	   p(c(x1)) -> c(p(x1))
8.35/3.03	   m(c(x1)) -> c(m(x1))
8.35/3.03	   n(c(x1)) -> c(n(x1))
8.35/3.03	   o(c(x1)) -> c(o(x1))
8.35/3.03	   o(c(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(5) RelTRSRRRProof (EQUIVALENT)
8.35/3.03	We used the following monotonic ordering for rule removal:
8.35/3.03	Polynomial interpretation [POLO]:
8.35/3.03	
8.35/3.03	   POL(c(x_1)) = x_1
8.35/3.03	   POL(m(x_1)) = x_1
8.35/3.03	   POL(n(x_1)) = x_1
8.35/3.03	   POL(o(x_1)) = 1 + x_1
8.35/3.03	   POL(p(x_1)) = x_1
8.35/3.03	   POL(s(x_1)) = x_1
8.35/3.03	   POL(t(x_1)) = x_1
8.35/3.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.35/3.03	Rules from R:
8.35/3.03	
8.35/3.03	   s(o(x1)) -> s(x1)
8.35/3.03	Rules from S:
8.35/3.03	none
8.35/3.03	
8.35/3.03	
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(6)
8.35/3.03	Obligation:
8.35/3.03	Relative term rewrite system:
8.35/3.03	The relative TRS consists of the following R rules:
8.35/3.03	
8.35/3.03	   s(n(x1)) -> s(x1)
8.35/3.03	
8.35/3.03	The relative TRS consists of the following S rules:
8.35/3.03	
8.35/3.03	   t(x1) -> n(c(t(x1)))
8.35/3.03	   n(o(x1)) -> o(n(x1))
8.35/3.03	   n(p(x1)) -> p(m(x1))
8.35/3.03	   m(p(x1)) -> p(m(x1))
8.35/3.03	   m(o(x1)) -> o(n(x1))
8.35/3.03	   p(n(x1)) -> n(p(x1))
8.35/3.03	   p(c(x1)) -> c(p(x1))
8.35/3.03	   m(c(x1)) -> c(m(x1))
8.35/3.03	   n(c(x1)) -> c(n(x1))
8.35/3.03	   o(c(x1)) -> c(o(x1))
8.35/3.03	   o(c(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(7) RelTRSRRRProof (EQUIVALENT)
8.35/3.03	We used the following monotonic ordering for rule removal:
8.35/3.03	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(s(x_1)) =  	[[2], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(n(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(t(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(o(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(p(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	   <<<
8.35/3.03	 POL(m(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.35/3.03	>>>
8.35/3.03	
8.35/3.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.35/3.03	Rules from R:
8.35/3.03	
8.35/3.03	   s(n(x1)) -> s(x1)
8.35/3.03	Rules from S:
8.35/3.03	none
8.35/3.03	
8.35/3.03	
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(8)
8.35/3.03	Obligation:
8.35/3.03	Relative term rewrite system:
8.35/3.03	R is empty.
8.35/3.03	The relative TRS consists of the following S rules:
8.35/3.03	
8.35/3.03	   t(x1) -> n(c(t(x1)))
8.35/3.03	   n(o(x1)) -> o(n(x1))
8.35/3.03	   n(p(x1)) -> p(m(x1))
8.35/3.03	   m(p(x1)) -> p(m(x1))
8.35/3.03	   m(o(x1)) -> o(n(x1))
8.35/3.03	   p(n(x1)) -> n(p(x1))
8.35/3.03	   p(c(x1)) -> c(p(x1))
8.35/3.03	   m(c(x1)) -> c(m(x1))
8.35/3.03	   n(c(x1)) -> c(n(x1))
8.35/3.03	   o(c(x1)) -> c(o(x1))
8.35/3.03	   o(c(x1)) -> o(x1)
8.35/3.03	
8.35/3.03	
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(9) RIsEmptyProof (EQUIVALENT)
8.35/3.03	The TRS R is empty. Hence, termination is trivially proven.
8.35/3.03	----------------------------------------
8.35/3.03	
8.35/3.03	(10)
8.35/3.03	YES
8.54/3.11	EOF