9.21/3.15	YES
9.21/3.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
9.21/3.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.21/3.16	
9.21/3.16	
9.21/3.16	Termination of the given RelTRS could be proven:
9.21/3.16	
9.21/3.16	(0) RelTRS
9.21/3.16	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
9.21/3.16	(2) RelTRS
9.21/3.16	(3) RelTRSRRRProof [EQUIVALENT, 34 ms]
9.21/3.16	(4) RelTRS
9.21/3.16	(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
9.21/3.16	(6) RelTRS
9.21/3.16	(7) RelTRSRRRProof [EQUIVALENT, 42 ms]
9.21/3.16	(8) RelTRS
9.21/3.16	(9) RelTRSRRRProof [EQUIVALENT, 54 ms]
9.21/3.16	(10) RelTRS
9.21/3.16	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
9.21/3.16	(12) YES
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(0)
9.21/3.16	Obligation:
9.21/3.16	Relative term rewrite system:
9.21/3.16	The relative TRS consists of the following R rules:
9.21/3.16	
9.21/3.16	   p(s(x1)) -> s(x1)
9.21/3.16	   n(s(x1)) -> s(x1)
9.21/3.16	   o(s(x1)) -> s(x1)
9.21/3.16	   o(n(x1)) -> n(o(x1))
9.21/3.16	   o(m(x1)) -> n(o(x1))
9.21/3.16	   n(o(p(x1))) -> o(n(x1))
9.21/3.16	
9.21/3.16	The relative TRS consists of the following S rules:
9.21/3.16	
9.21/3.16	   t(x1) -> t(c(n(x1)))
9.21/3.16	   p(n(x1)) -> m(p(x1))
9.21/3.16	   p(m(x1)) -> m(p(x1))
9.21/3.16	   n(p(x1)) -> p(n(x1))
9.21/3.16	   c(p(x1)) -> p(c(x1))
9.21/3.16	   c(m(x1)) -> m(c(x1))
9.21/3.16	   c(n(x1)) -> n(c(x1))
9.21/3.16	   c(o(x1)) -> o(c(x1))
9.21/3.16	   c(o(x1)) -> o(x1)
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(1) RelTRS Reverse (EQUIVALENT)
9.21/3.16	We have reversed the following relative TRS [REVERSE]:
9.21/3.16	The set of rules R is 
9.21/3.16	   p(s(x1)) -> s(x1)
9.21/3.16	   n(s(x1)) -> s(x1)
9.21/3.16	   o(s(x1)) -> s(x1)
9.21/3.16	   o(n(x1)) -> n(o(x1))
9.21/3.16	   o(m(x1)) -> n(o(x1))
9.21/3.16	   n(o(p(x1))) -> o(n(x1))
9.21/3.16	
9.21/3.16	The set of rules S is 
9.21/3.16	   t(x1) -> t(c(n(x1)))
9.21/3.16	   p(n(x1)) -> m(p(x1))
9.21/3.16	   p(m(x1)) -> m(p(x1))
9.21/3.16	   n(p(x1)) -> p(n(x1))
9.21/3.16	   c(p(x1)) -> p(c(x1))
9.21/3.16	   c(m(x1)) -> m(c(x1))
9.21/3.16	   c(n(x1)) -> n(c(x1))
9.21/3.16	   c(o(x1)) -> o(c(x1))
9.21/3.16	   c(o(x1)) -> o(x1)
9.21/3.16	
9.21/3.16	We have obtained the following relative TRS:
9.21/3.16	The set of rules R is 
9.21/3.16	   s(p(x1)) -> s(x1)
9.21/3.16	   s(n(x1)) -> s(x1)
9.21/3.16	   s(o(x1)) -> s(x1)
9.21/3.16	   n(o(x1)) -> o(n(x1))
9.21/3.16	   m(o(x1)) -> o(n(x1))
9.21/3.16	   p(o(n(x1))) -> n(o(x1))
9.21/3.16	
9.21/3.16	The set of rules S is 
9.21/3.16	   t(x1) -> n(c(t(x1)))
9.21/3.16	   n(p(x1)) -> p(m(x1))
9.21/3.16	   m(p(x1)) -> p(m(x1))
9.21/3.16	   p(n(x1)) -> n(p(x1))
9.21/3.16	   p(c(x1)) -> c(p(x1))
9.21/3.16	   m(c(x1)) -> c(m(x1))
9.21/3.16	   n(c(x1)) -> c(n(x1))
9.21/3.16	   o(c(x1)) -> c(o(x1))
9.21/3.16	   o(c(x1)) -> o(x1)
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(2)
9.21/3.16	Obligation:
9.21/3.16	Relative term rewrite system:
9.21/3.16	The relative TRS consists of the following R rules:
9.21/3.16	
9.21/3.16	   s(p(x1)) -> s(x1)
9.21/3.16	   s(n(x1)) -> s(x1)
9.21/3.16	   s(o(x1)) -> s(x1)
9.21/3.16	   n(o(x1)) -> o(n(x1))
9.21/3.16	   m(o(x1)) -> o(n(x1))
9.21/3.16	   p(o(n(x1))) -> n(o(x1))
9.21/3.16	
9.21/3.16	The relative TRS consists of the following S rules:
9.21/3.16	
9.21/3.16	   t(x1) -> n(c(t(x1)))
9.21/3.16	   n(p(x1)) -> p(m(x1))
9.21/3.16	   m(p(x1)) -> p(m(x1))
9.21/3.16	   p(n(x1)) -> n(p(x1))
9.21/3.16	   p(c(x1)) -> c(p(x1))
9.21/3.16	   m(c(x1)) -> c(m(x1))
9.21/3.16	   n(c(x1)) -> c(n(x1))
9.21/3.16	   o(c(x1)) -> c(o(x1))
9.21/3.16	   o(c(x1)) -> o(x1)
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(3) RelTRSRRRProof (EQUIVALENT)
9.21/3.16	We used the following monotonic ordering for rule removal:
9.21/3.16	Polynomial interpretation [POLO]:
9.21/3.16	
9.21/3.16	   POL(c(x_1)) = x_1
9.21/3.16	   POL(m(x_1)) = x_1
9.21/3.16	   POL(n(x_1)) = x_1
9.21/3.16	   POL(o(x_1)) = x_1
9.21/3.16	   POL(p(x_1)) = 1 + x_1
9.21/3.16	   POL(s(x_1)) = x_1
9.21/3.16	   POL(t(x_1)) = x_1
9.21/3.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.21/3.16	Rules from R:
9.21/3.16	
9.21/3.16	   s(p(x1)) -> s(x1)
9.21/3.16	   p(o(n(x1))) -> n(o(x1))
9.21/3.16	Rules from S:
9.21/3.16	none
9.21/3.16	
9.21/3.16	
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(4)
9.21/3.16	Obligation:
9.21/3.16	Relative term rewrite system:
9.21/3.16	The relative TRS consists of the following R rules:
9.21/3.16	
9.21/3.16	   s(n(x1)) -> s(x1)
9.21/3.16	   s(o(x1)) -> s(x1)
9.21/3.16	   n(o(x1)) -> o(n(x1))
9.21/3.16	   m(o(x1)) -> o(n(x1))
9.21/3.16	
9.21/3.16	The relative TRS consists of the following S rules:
9.21/3.16	
9.21/3.16	   t(x1) -> n(c(t(x1)))
9.21/3.16	   n(p(x1)) -> p(m(x1))
9.21/3.16	   m(p(x1)) -> p(m(x1))
9.21/3.16	   p(n(x1)) -> n(p(x1))
9.21/3.16	   p(c(x1)) -> c(p(x1))
9.21/3.16	   m(c(x1)) -> c(m(x1))
9.21/3.16	   n(c(x1)) -> c(n(x1))
9.21/3.16	   o(c(x1)) -> c(o(x1))
9.21/3.16	   o(c(x1)) -> o(x1)
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(5) RelTRSRRRProof (EQUIVALENT)
9.21/3.16	We used the following monotonic ordering for rule removal:
9.21/3.16	Polynomial interpretation [POLO]:
9.21/3.16	
9.21/3.16	   POL(c(x_1)) = x_1
9.21/3.16	   POL(m(x_1)) = x_1
9.21/3.16	   POL(n(x_1)) = x_1
9.21/3.16	   POL(o(x_1)) = 1 + x_1
9.21/3.16	   POL(p(x_1)) = x_1
9.21/3.16	   POL(s(x_1)) = x_1
9.21/3.16	   POL(t(x_1)) = x_1
9.21/3.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.21/3.16	Rules from R:
9.21/3.16	
9.21/3.16	   s(o(x1)) -> s(x1)
9.21/3.16	Rules from S:
9.21/3.16	none
9.21/3.16	
9.21/3.16	
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(6)
9.21/3.16	Obligation:
9.21/3.16	Relative term rewrite system:
9.21/3.16	The relative TRS consists of the following R rules:
9.21/3.16	
9.21/3.16	   s(n(x1)) -> s(x1)
9.21/3.16	   n(o(x1)) -> o(n(x1))
9.21/3.16	   m(o(x1)) -> o(n(x1))
9.21/3.16	
9.21/3.16	The relative TRS consists of the following S rules:
9.21/3.16	
9.21/3.16	   t(x1) -> n(c(t(x1)))
9.21/3.16	   n(p(x1)) -> p(m(x1))
9.21/3.16	   m(p(x1)) -> p(m(x1))
9.21/3.16	   p(n(x1)) -> n(p(x1))
9.21/3.16	   p(c(x1)) -> c(p(x1))
9.21/3.16	   m(c(x1)) -> c(m(x1))
9.21/3.16	   n(c(x1)) -> c(n(x1))
9.21/3.16	   o(c(x1)) -> c(o(x1))
9.21/3.16	   o(c(x1)) -> o(x1)
9.21/3.16	
9.21/3.16	
9.21/3.16	----------------------------------------
9.21/3.16	
9.21/3.16	(7) RelTRSRRRProof (EQUIVALENT)
9.21/3.16	We used the following monotonic ordering for rule removal:
9.21/3.16	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
9.21/3.16	
9.21/3.16	   <<<
9.21/3.16	 POL(s(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
9.21/3.16	>>>
9.21/3.16	
9.21/3.16	   <<<
9.21/3.16	 POL(n(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
9.21/3.16	>>>
9.21/3.16	
9.21/3.16	   <<<
9.21/3.16	 POL(o(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
9.21/3.16	>>>
9.21/3.16	
9.21/3.16	   <<<
9.21/3.16	 POL(m(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
9.21/3.16	>>>
9.21/3.16	
9.21/3.16	   <<<
9.21/3.16	 POL(t(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
9.21/3.16	>>>
9.21/3.16	
9.21/3.16	   <<<
9.21/3.16	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(p(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.33/3.16	Rules from R:
9.33/3.16	
9.33/3.16	   s(n(x1)) -> s(x1)
9.33/3.16	Rules from S:
9.33/3.16	none
9.33/3.16	
9.33/3.16	
9.33/3.16	
9.33/3.16	
9.33/3.16	----------------------------------------
9.33/3.16	
9.33/3.16	(8)
9.33/3.16	Obligation:
9.33/3.16	Relative term rewrite system:
9.33/3.16	The relative TRS consists of the following R rules:
9.33/3.16	
9.33/3.16	   n(o(x1)) -> o(n(x1))
9.33/3.16	   m(o(x1)) -> o(n(x1))
9.33/3.16	
9.33/3.16	The relative TRS consists of the following S rules:
9.33/3.16	
9.33/3.16	   t(x1) -> n(c(t(x1)))
9.33/3.16	   n(p(x1)) -> p(m(x1))
9.33/3.16	   m(p(x1)) -> p(m(x1))
9.33/3.16	   p(n(x1)) -> n(p(x1))
9.33/3.16	   p(c(x1)) -> c(p(x1))
9.33/3.16	   m(c(x1)) -> c(m(x1))
9.33/3.16	   n(c(x1)) -> c(n(x1))
9.33/3.16	   o(c(x1)) -> c(o(x1))
9.33/3.16	   o(c(x1)) -> o(x1)
9.33/3.16	
9.33/3.16	
9.33/3.16	----------------------------------------
9.33/3.16	
9.33/3.16	(9) RelTRSRRRProof (EQUIVALENT)
9.33/3.16	We used the following monotonic ordering for rule removal:
9.33/3.16	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(n(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(o(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(m(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(t(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	   <<<
9.33/3.16	 POL(p(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
9.33/3.16	>>>
9.33/3.16	
9.33/3.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.33/3.16	Rules from R:
9.33/3.16	
9.33/3.16	   n(o(x1)) -> o(n(x1))
9.33/3.16	   m(o(x1)) -> o(n(x1))
9.33/3.16	Rules from S:
9.33/3.16	none
9.33/3.16	
9.33/3.16	
9.33/3.16	
9.33/3.16	
9.33/3.16	----------------------------------------
9.33/3.16	
9.33/3.16	(10)
9.33/3.16	Obligation:
9.33/3.16	Relative term rewrite system:
9.33/3.16	R is empty.
9.33/3.16	The relative TRS consists of the following S rules:
9.33/3.16	
9.33/3.16	   t(x1) -> n(c(t(x1)))
9.33/3.16	   n(p(x1)) -> p(m(x1))
9.33/3.16	   m(p(x1)) -> p(m(x1))
9.33/3.16	   p(n(x1)) -> n(p(x1))
9.33/3.16	   p(c(x1)) -> c(p(x1))
9.33/3.16	   m(c(x1)) -> c(m(x1))
9.33/3.16	   n(c(x1)) -> c(n(x1))
9.33/3.16	   o(c(x1)) -> c(o(x1))
9.33/3.16	   o(c(x1)) -> o(x1)
9.33/3.16	
9.33/3.16	
9.33/3.16	----------------------------------------
9.33/3.16	
9.33/3.16	(11) RIsEmptyProof (EQUIVALENT)
9.33/3.16	The TRS R is empty. Hence, termination is trivially proven.
9.33/3.16	----------------------------------------
9.33/3.16	
9.33/3.16	(12)
9.33/3.16	YES
9.33/3.21	EOF