/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 42 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 5 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   f(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set
false: 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:

   f(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set
false: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   if(false, X, Y) -> Y




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

   f(X) -> if(X, c, f(true))
   if(true, X, Y) -> X

The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set

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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(X) -> if(X, c, f(true))
   if(true, X, Y) -> X

The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   f(X) -> if(X, c, f(true))




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

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

   if(true, X, Y) -> X

The replacement map contains the following entries:

if: {1, 2}
true: empty set

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

   if(true, X, Y) -> X

The replacement map contains the following entries:

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

   <<<
 POL(if(x_1, x_2, x_3)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2 	 +  	[[1, 1], [0, 1]] 	* 	x_3
>>>

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

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

   if(true, X, Y) -> X




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(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(7) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(8)
YES