7.90/2.87	YES
7.90/2.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.90/2.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.90/2.88	
7.90/2.88	
7.90/2.88	Termination of the given RelTRS could be proven:
7.90/2.88	
7.90/2.88	(0) RelTRS
7.90/2.88	(1) RelTRSRRRProof [EQUIVALENT, 32 ms]
7.90/2.88	(2) RelTRS
7.90/2.88	(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
7.90/2.88	(4) RelTRS
7.90/2.88	(5) RelTRSRRRProof [EQUIVALENT, 52 ms]
7.90/2.88	(6) RelTRS
7.90/2.88	(7) RelTRSRRRProof [EQUIVALENT, 21 ms]
7.90/2.88	(8) RelTRS
7.90/2.88	(9) RelTRSRRRProof [EQUIVALENT, 24 ms]
7.90/2.88	(10) RelTRS
7.90/2.88	(11) RelTRSRRRProof [EQUIVALENT, 12 ms]
7.90/2.88	(12) RelTRS
7.90/2.88	(13) RIsEmptyProof [EQUIVALENT, 2 ms]
7.90/2.88	(14) YES
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(0)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	The relative TRS consists of the following R rules:
7.90/2.88	
7.90/2.88	   app(nil, k) -> k
7.90/2.88	   app(l, nil) -> l
7.90/2.88	   app(cons(x, l), k) -> cons(x, app(l, k))
7.90/2.88	   sum(cons(x, nil)) -> cons(x, nil)
7.90/2.88	   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	   plus(0, y) -> y
7.90/2.88	   plus(s(x), y) -> s(plus(x, y))
7.90/2.88	
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(1) RelTRSRRRProof (EQUIVALENT)
7.90/2.88	We used the following monotonic ordering for rule removal:
7.90/2.88	Polynomial interpretation [POLO]:
7.90/2.88	
7.90/2.88	   POL(0) = 0
7.90/2.88	   POL(app(x_1, x_2)) = x_1 + x_2
7.90/2.88	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
7.90/2.88	   POL(nil) = 1
7.90/2.88	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
7.90/2.88	   POL(s(x_1)) = x_1
7.90/2.88	   POL(sum(x_1)) = x_1
7.90/2.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.90/2.88	Rules from R:
7.90/2.88	
7.90/2.88	   app(nil, k) -> k
7.90/2.88	   app(l, nil) -> l
7.90/2.88	   plus(0, y) -> y
7.90/2.88	Rules from S:
7.90/2.88	none
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(2)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	The relative TRS consists of the following R rules:
7.90/2.88	
7.90/2.88	   app(cons(x, l), k) -> cons(x, app(l, k))
7.90/2.88	   sum(cons(x, nil)) -> cons(x, nil)
7.90/2.88	   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	   plus(s(x), y) -> s(plus(x, y))
7.90/2.88	
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(3) RelTRSRRRProof (EQUIVALENT)
7.90/2.88	We used the following monotonic ordering for rule removal:
7.90/2.88	Polynomial interpretation [POLO]:
7.90/2.88	
7.90/2.88	   POL(app(x_1, x_2)) = x_1 + x_2
7.90/2.88	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
7.90/2.88	   POL(nil) = 0
7.90/2.88	   POL(plus(x_1, x_2)) = x_1 + x_2
7.90/2.88	   POL(s(x_1)) = x_1
7.90/2.88	   POL(sum(x_1)) = x_1
7.90/2.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.90/2.88	Rules from R:
7.90/2.88	
7.90/2.88	   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
7.90/2.88	Rules from S:
7.90/2.88	none
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(4)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	The relative TRS consists of the following R rules:
7.90/2.88	
7.90/2.88	   app(cons(x, l), k) -> cons(x, app(l, k))
7.90/2.88	   sum(cons(x, nil)) -> cons(x, nil)
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	   plus(s(x), y) -> s(plus(x, y))
7.90/2.88	
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(5) RelTRSRRRProof (EQUIVALENT)
7.90/2.88	We used the following monotonic ordering for rule removal:
7.90/2.88	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(app(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(sum(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(nil) =  	[[0], [0]]
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(plus(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(s(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.90/2.88	Rules from R:
7.90/2.88	
7.90/2.88	   plus(s(x), y) -> s(plus(x, y))
7.90/2.88	Rules from S:
7.90/2.88	none
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(6)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	The relative TRS consists of the following R rules:
7.90/2.88	
7.90/2.88	   app(cons(x, l), k) -> cons(x, app(l, k))
7.90/2.88	   sum(cons(x, nil)) -> cons(x, nil)
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(7) RelTRSRRRProof (EQUIVALENT)
7.90/2.88	We used the following monotonic ordering for rule removal:
7.90/2.88	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(app(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(cons(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(sum(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(nil) =  	[[0], [0]]
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.90/2.88	Rules from R:
7.90/2.88	
7.90/2.88	   app(cons(x, l), k) -> cons(x, app(l, k))
7.90/2.88	Rules from S:
7.90/2.88	none
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(8)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	The relative TRS consists of the following R rules:
7.90/2.88	
7.90/2.88	   sum(cons(x, nil)) -> cons(x, nil)
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(9) RelTRSRRRProof (EQUIVALENT)
7.90/2.88	We used the following monotonic ordering for rule removal:
7.90/2.88	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(sum(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(cons(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(nil) =  	[[0], [0]]
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(app(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.90/2.88	Rules from R:
7.90/2.88	
7.90/2.88	   sum(cons(x, nil)) -> cons(x, nil)
7.90/2.88	Rules from S:
7.90/2.88	none
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(10)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	The relative TRS consists of the following R rules:
7.90/2.88	
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(11) RelTRSRRRProof (EQUIVALENT)
7.90/2.88	We used the following monotonic ordering for rule removal:
7.90/2.88	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(sum(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(app(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	   <<<
7.90/2.88	 POL(cons(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
7.90/2.88	>>>
7.90/2.88	
7.90/2.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.90/2.88	Rules from R:
7.90/2.88	
7.90/2.88	   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
7.90/2.88	Rules from S:
7.90/2.88	none
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(12)
7.90/2.88	Obligation:
7.90/2.88	Relative term rewrite system:
7.90/2.88	R is empty.
7.90/2.88	The relative TRS consists of the following S rules:
7.90/2.88	
7.90/2.88	   cons(x, cons(y, l)) -> cons(y, cons(x, l))
7.90/2.88	
7.90/2.88	
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(13) RIsEmptyProof (EQUIVALENT)
7.90/2.88	The TRS R is empty. Hence, termination is trivially proven.
7.90/2.88	----------------------------------------
7.90/2.88	
7.90/2.88	(14)
7.90/2.88	YES
8.33/3.04	EOF