3.65/1.93	YES
3.65/1.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.65/1.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.65/1.94	
3.65/1.94	
3.65/1.94	Termination of the given CSR could be proven:
3.65/1.94	
3.65/1.94	(0) CSR
3.65/1.94	(1) CSRRRRProof [EQUIVALENT, 81 ms]
3.65/1.94	(2) CSR
3.65/1.94	(3) CSRRRRProof [EQUIVALENT, 6 ms]
3.65/1.94	(4) CSR
3.65/1.94	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.65/1.94	(6) CSR
3.65/1.94	(7) CSRRRRProof [EQUIVALENT, 8 ms]
3.65/1.94	(8) CSR
3.65/1.94	(9) CSRRRRProof [EQUIVALENT, 0 ms]
3.65/1.94	(10) CSR
3.65/1.94	(11) CSRRRRProof [EQUIVALENT, 0 ms]
3.65/1.94	(12) CSR
3.65/1.94	(13) RisEmptyProof [EQUIVALENT, 0 ms]
3.65/1.94	(14) YES
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(0)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	   hd(cons(X, Y)) -> X
3.65/1.94	   tl(cons(X, Y)) -> Y
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	nats: empty set
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	hd: {1}
3.65/1.94	tl: {1}
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(1) CSRRRRProof (EQUIVALENT)
3.65/1.94	The following CSR is given: Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	   hd(cons(X, Y)) -> X
3.65/1.94	   tl(cons(X, Y)) -> Y
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	nats: empty set
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	hd: {1}
3.65/1.94	tl: {1}
3.65/1.94	Used ordering:
3.65/1.94	Polynomial interpretation [POLO]:
3.65/1.94	
3.65/1.94	   POL(0) = 0
3.65/1.94	   POL(adx(x_1)) = 1 + x_1
3.65/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
3.65/1.94	   POL(hd(x_1)) = 1 + x_1
3.65/1.94	   POL(incr(x_1)) = x_1
3.65/1.94	   POL(nats) = 1
3.65/1.94	   POL(s(x_1)) = 0
3.65/1.94	   POL(tl(x_1)) = x_1
3.65/1.94	   POL(zeros) = 0
3.65/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.94	
3.65/1.94	   hd(cons(X, Y)) -> X
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(2)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	   tl(cons(X, Y)) -> Y
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	nats: empty set
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	tl: {1}
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(3) CSRRRRProof (EQUIVALENT)
3.65/1.94	The following CSR is given: Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	   tl(cons(X, Y)) -> Y
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	nats: empty set
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	tl: {1}
3.65/1.94	Used ordering:
3.65/1.94	Polynomial interpretation [POLO]:
3.65/1.94	
3.65/1.94	   POL(0) = 0
3.65/1.94	   POL(adx(x_1)) = 1 + x_1
3.65/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
3.65/1.94	   POL(incr(x_1)) = x_1
3.65/1.94	   POL(nats) = 1
3.65/1.94	   POL(s(x_1)) = x_1
3.65/1.94	   POL(tl(x_1)) = 1 + x_1
3.65/1.94	   POL(zeros) = 0
3.65/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.94	
3.65/1.94	   tl(cons(X, Y)) -> Y
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(4)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	nats: empty set
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(5) CSRRRRProof (EQUIVALENT)
3.65/1.94	The following CSR is given: Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	nats: empty set
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	Used ordering:
3.65/1.94	Polynomial interpretation [POLO]:
3.65/1.94	
3.65/1.94	   POL(0) = 0
3.65/1.94	   POL(adx(x_1)) = x_1
3.65/1.94	   POL(cons(x_1, x_2)) = x_1 + x_2
3.65/1.94	   POL(incr(x_1)) = x_1
3.65/1.94	   POL(nats) = 1
3.65/1.94	   POL(s(x_1)) = x_1
3.65/1.94	   POL(zeros) = 0
3.65/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.94	
3.65/1.94	   nats -> adx(zeros)
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(6)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(7) CSRRRRProof (EQUIVALENT)
3.65/1.94	The following CSR is given: Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	adx: {1}
3.65/1.94	zeros: empty set
3.65/1.94	cons: empty set
3.65/1.94	0: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	Used ordering:
3.65/1.94	Polynomial interpretation [POLO]:
3.65/1.94	
3.65/1.94	   POL(0) = 2
3.65/1.94	   POL(adx(x_1)) = x_1
3.65/1.94	   POL(cons(x_1, x_2)) = 0
3.65/1.94	   POL(incr(x_1)) = 2*x_1
3.65/1.94	   POL(s(x_1)) = 2 + x_1
3.65/1.94	   POL(zeros) = 1
3.65/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.94	
3.65/1.94	   zeros -> cons(0, zeros)
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(8)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	adx: {1}
3.65/1.94	cons: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(9) CSRRRRProof (EQUIVALENT)
3.65/1.94	The following CSR is given: Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	adx: {1}
3.65/1.94	cons: empty set
3.65/1.94	incr: {1}
3.65/1.94	s: empty set
3.65/1.94	Used ordering:
3.65/1.94	Polynomial interpretation [POLO]:
3.65/1.94	
3.65/1.94	   POL(adx(x_1)) = 1 + x_1
3.65/1.94	   POL(cons(x_1, x_2)) = x_1
3.65/1.94	   POL(incr(x_1)) = 1 + x_1
3.65/1.94	   POL(s(x_1)) = x_1
3.65/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.94	
3.65/1.94	   incr(cons(X, Y)) -> cons(s(X), incr(Y))
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(10)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	adx: {1}
3.65/1.94	cons: empty set
3.65/1.94	incr: {1}
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(11) CSRRRRProof (EQUIVALENT)
3.65/1.94	The following CSR is given: Context-sensitive rewrite system:
3.65/1.94	The TRS R consists of the following rules:
3.65/1.94	
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	The replacement map contains the following entries:
3.65/1.94	
3.65/1.94	adx: {1}
3.65/1.94	cons: empty set
3.65/1.94	incr: {1}
3.65/1.94	Used ordering:
3.65/1.94	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.65/1.94	
3.65/1.94	   <<<
3.65/1.94	 POL(adx(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
3.65/1.94	>>>
3.65/1.94	
3.65/1.94	   <<<
3.65/1.94	 POL(cons(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
3.65/1.94	>>>
3.65/1.94	
3.65/1.94	   <<<
3.65/1.94	 POL(incr(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.65/1.94	>>>
3.65/1.94	
3.65/1.94	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.94	
3.65/1.94	   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(12)
3.65/1.94	Obligation:
3.65/1.94	Context-sensitive rewrite system:
3.65/1.94	R is empty.
3.65/1.94	
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(13) RisEmptyProof (EQUIVALENT)
3.65/1.94	The CSR R is empty. Hence, termination is trivially proven.
3.65/1.94	----------------------------------------
3.65/1.94	
3.65/1.94	(14)
3.65/1.94	YES
3.65/1.96	EOF