/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 52 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 2 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(10) QCSDP
(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(g_1(x))) -> a_0
   f_1(x, y) -> x
   f_1(x, i_0(x)) -> f_1(x, x)
   g_1(x) -> i_0(x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
a_0: 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_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(g_1(x))) -> a_0
   f_1(x, y) -> x
   f_1(x, i_0(x)) -> f_1(x, x)
   g_1(x) -> i_0(x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
a_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_0) = 2
   POL(f_1(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(g_1(x_1)) = x_1
   POL(i_0(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_1(x, y) -> x




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

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(g_1(x))) -> a_0
   f_1(x, i_0(x)) -> f_1(x, x)
   g_1(x) -> i_0(x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
a_0: 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_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(g_1(x))) -> a_0
   f_1(x, i_0(x)) -> f_1(x, x)
   g_1(x) -> i_0(x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
a_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_0) = 1
   POL(f_1(x_1, x_2)) = 1 + x_1
   POL(g_1(x_1)) = 1 + x_1
   POL(i_0(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g_1(x) -> i_0(x)




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

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

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(g_1(x))) -> a_0
   f_1(x, i_0(x)) -> f_1(x, x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
a_0: 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:

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(g_1(x))) -> a_0
   f_1(x, i_0(x)) -> f_1(x, x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
a_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_0) = 0
   POL(f_1(x_1, x_2)) = 1
   POL(g_1(x_1)) = 1 + x_1
   POL(i_0(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:

   f_1(x, i_0(g_1(x))) -> a_0




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

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

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(x)) -> f_1(x, x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set

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

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

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
   f_1(x, i_0(x)) -> f_1(x, x)

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(f_1(x_1, x_2)) =  	[[1]] 	 +  	[[1, 0]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2
>>>

   <<<
 POL(i_0(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(g_1(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [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_1(x, i_0(x)) -> f_1(x, x)




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(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))

The replacement map contains the following entries:

f_1: empty set
i_0: {1}
g_1: empty set

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(9) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
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(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {i_0_1} are replacing on all positions.
The symbols in {f_1_2, g_1_1, F_1_2} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   F_1(x, x) -> F_1(i_0(x), g_1(g_1(x)))

The TRS R consists of the following rules:

   f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))

Q is empty.

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(11) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F_1(z0, z0) -> F_1(i_0(z0), g_1(g_1(z0))) and F_1(x0, x0) -> F_1(i_0(x0), g_1(g_1(x0))) form no chain, because ECap^mu(F_1(i_0(z0), g_1(g_1(z0)))) = F_1(i_0(z0), g_1(g_1(z0))) does not unify with F_1(x0, x0). 
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(12)
TRUE