/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 49 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:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(0)) -> mark(cons(0, f(s(0))))
   active(f(s(0))) -> mark(f(p(s(0))))
   active(p(s(X))) -> mark(X)
   active(f(X)) -> f(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(p(X)) -> p(active(X))
   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   p(mark(X)) -> mark(p(X))
   proper(f(X)) -> f(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(p(X)) -> p(proper(X))
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   p(ok(X)) -> ok(p(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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(1) QTRSToCSRProof (EQUIVALENT)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(0)) -> mark(cons(0, f(s(0))))
   active(f(s(0))) -> mark(f(p(s(0))))
   active(p(s(X))) -> mark(X)
   active(f(X)) -> f(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(p(X)) -> p(active(X))
   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   p(mark(X)) -> mark(p(X))
   proper(f(X)) -> f(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(p(X)) -> p(proper(X))
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   p(ok(X)) -> ok(p(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

f: {1}
0: empty set
cons: {1}
s: {1}
p: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
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(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(X)) -> X

The replacement map contains the following entries:

f: {1}
0: empty set
cons: {1}
s: {1}
p: {1}

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

   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(X)) -> X

The replacement map contains the following entries:

f: {1}
0: empty set
cons: {1}
s: {1}
p: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = x_1
   POL(f(x_1)) = 1 + x_1
   POL(p(x_1)) = 2*x_1
   POL(s(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:

   f(0) -> cons(0, f(s(0)))




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

   f(s(0)) -> f(p(s(0)))
   p(s(X)) -> X

The replacement map contains the following entries:

f: {1}
0: empty set
s: {1}
p: {1}

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

   f(s(0)) -> f(p(s(0)))
   p(s(X)) -> X

The replacement map contains the following entries:

f: {1}
0: empty set
s: {1}
p: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(f(x_1)) = x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   p(s(X)) -> X




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

   f(s(0)) -> f(p(s(0)))

The replacement map contains the following entries:

f: {1}
0: empty set
s: {1}
p: {1}

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

   f(s(0)) -> f(p(s(0)))

The replacement map contains the following entries:

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

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

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

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

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

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

   f(s(0)) -> f(p(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