/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 112 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:

   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
   sieve(cons(0, Y)) -> cons(0, sieve(Y))
   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
   nats(N) -> cons(N, nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

filter: {1, 2, 3}
cons: {1}
0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set

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

   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
   sieve(cons(0, Y)) -> cons(0, sieve(Y))
   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
   nats(N) -> cons(N, nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

filter: {1, 2, 3}
cons: {1}
0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = x_1
   POL(filter(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(nats(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(sieve(x_1)) = 1 + x_1
   POL(zprimes) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   sieve(cons(0, Y)) -> cons(0, sieve(Y))
   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))




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

   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
   nats(N) -> cons(N, nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

filter: {1, 2, 3}
cons: {1}
0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set

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

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

   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
   nats(N) -> cons(N, nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

filter: {1, 2, 3}
cons: {1}
0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 1 + x_1
   POL(filter(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(nats(x_1)) = 1 + x_1
   POL(s(x_1)) = x_1
   POL(sieve(x_1)) = x_1
   POL(zprimes) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))




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

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

   nats(N) -> cons(N, nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

cons: {1}
0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set

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

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

   nats(N) -> cons(N, nats(s(N)))
   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

cons: {1}
0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   nats(N) -> cons(N, nats(s(N)))




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

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

   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set

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

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

   zprimes -> sieve(nats(s(s(0))))

The replacement map contains the following entries:

0: empty set
s: {1}
sieve: {1}
nats: {1}
zprimes: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

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

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

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

   zprimes -> sieve(nats(s(s(0))))




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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