/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 98 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 1 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:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

minus: empty set
0: empty set
s: {1}
geq: empty set
true: empty set
false: empty set
div: {1}
if: {1}

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

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

minus: empty set
0: empty set
s: {1}
geq: empty set
true: empty set
false: empty set
div: {1}
if: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(div(x_1, x_2)) = 2*x_1
   POL(false) = 0
   POL(geq(x_1, x_2)) = x_1
   POL(if(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(minus(x_1, x_2)) = 0
   POL(s(x_1)) = 1 + 2*x_1
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   geq(s(X), s(Y)) -> geq(X, Y)




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

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

minus: empty set
0: empty set
s: {1}
geq: empty set
true: empty set
false: empty set
div: {1}
if: {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:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

minus: empty set
0: empty set
s: {1}
geq: empty set
true: empty set
false: empty set
div: {1}
if: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(div(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(false) = 0
   POL(geq(x_1, x_2)) = 0
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(minus(x_1, x_2)) = 0
   POL(s(x_1)) = 1 + x_1
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)




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

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

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

minus: empty set
0: empty set
s: {1}
geq: empty set
true: empty set
false: empty set
if: {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:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

minus: empty set
0: empty set
s: {1}
geq: empty set
true: empty set
false: empty set
if: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(false) = 1
   POL(geq(x_1, x_2)) = 1 + x_1 + x_2
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(minus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = 1 + x_1
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   if(false, X, Y) -> Y




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

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

   minus(0, Y) -> 0
   if(true, X, Y) -> X

The replacement map contains the following entries:

minus: empty set
0: empty set
true: empty set
if: {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:

   minus(0, Y) -> 0
   if(true, X, Y) -> X

The replacement map contains the following entries:

minus: empty set
0: empty set
true: empty set
if: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(if(x_1, x_2, x_3)) = 2*x_1 + x_2
   POL(minus(x_1, x_2)) = 1 + 2*x_1
   POL(true) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   minus(0, Y) -> 0
   if(true, X, Y) -> X




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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