/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 84 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
head: {1}
tail: {1}

----------------------------------------

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
head: {1}
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(head(x_1)) = 1 + 2*x_1
   POL(incr(x_1)) = x_1
   POL(nats) = 0
   POL(odds) = 0
   POL(pairs) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   head(cons(X, XS)) -> X




----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
tail: {1}

----------------------------------------

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(incr(x_1)) = x_1
   POL(nats) = 0
   POL(odds) = 0
   POL(pairs) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = 1 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   tail(cons(X, XS)) -> XS




----------------------------------------

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}

----------------------------------------

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 2*x_1
   POL(incr(x_1)) = 1 + x_1
   POL(nats) = 1
   POL(odds) = 2
   POL(pairs) = 0
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   nats -> cons(0, incr(nats))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))




----------------------------------------

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   pairs -> cons(0, incr(odds))

The replacement map contains the following entries:

cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set

----------------------------------------

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   pairs -> cons(0, incr(odds))

The replacement map contains the following entries:

cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(pairs) =  	[[1]]
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1 	 +  	[[1, 0]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(incr(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(odds) =  	[[0], [1]]
>>>

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   pairs -> cons(0, incr(odds))




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(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(9) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(10)
YES