YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 54 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 4 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

   f_1(h_1(x)) -> f_1(i_1(x))
   h_1(x) -> f_1(h_1(x))
   *top*_0(f_1(i_1(x))) -> *top*_0(x)
   f_0(f_1(i_1(x))) -> f_1(x)
   f_1(i_1(x)) -> x
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
*top*_0: {1}
f_0: {1}

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

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

   f_1(h_1(x)) -> f_1(i_1(x))
   h_1(x) -> f_1(h_1(x))
   *top*_0(f_1(i_1(x))) -> *top*_0(x)
   f_0(f_1(i_1(x))) -> f_1(x)
   f_1(i_1(x)) -> x
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
*top*_0: {1}
f_0: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = x_1
   POL(f_0(x_1)) = 2 + x_1
   POL(f_1(x_1)) = x_1
   POL(h_1(x_1)) = 2*x_1
   POL(i_1(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(f_1(i_1(x))) -> f_1(x)




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

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

   f_1(h_1(x)) -> f_1(i_1(x))
   h_1(x) -> f_1(h_1(x))
   *top*_0(f_1(i_1(x))) -> *top*_0(x)
   f_1(i_1(x)) -> x
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
*top*_0: {1}

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

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

   f_1(h_1(x)) -> f_1(i_1(x))
   h_1(x) -> f_1(h_1(x))
   *top*_0(f_1(i_1(x))) -> *top*_0(x)
   f_1(i_1(x)) -> x
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
*top*_0: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = x_1
   POL(f_1(x_1)) = x_1
   POL(h_1(x_1)) = 1 + x_1
   POL(i_1(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:

   *top*_0(f_1(i_1(x))) -> *top*_0(x)
   f_1(i_1(x)) -> x




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

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

   f_1(h_1(x)) -> f_1(i_1(x))
   h_1(x) -> f_1(h_1(x))
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set

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

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

   f_1(h_1(x)) -> f_1(i_1(x))
   h_1(x) -> f_1(h_1(x))
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(f_1(x_1)) = 0
   POL(h_1(x_1)) = 1 + x_1
   POL(i_1(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:

   h_1(x) -> f_1(h_1(x))




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

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

   f_1(h_1(x)) -> f_1(i_1(x))
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_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(h_1(x)) -> f_1(i_1(x))
   i_1(x) -> h_1(x)

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(f_1(x_1)) = 0
   POL(h_1(x_1)) = 1 + x_1
   POL(i_1(x_1)) = 2 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   i_1(x) -> h_1(x)




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

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

   f_1(h_1(x)) -> f_1(i_1(x))

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set

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

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

   f_1(h_1(x)) -> f_1(i_1(x))

The replacement map contains the following entries:

f_1: empty set
h_1: empty set
i_1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(f_1(x_1)) = 2*x_1
   POL(h_1(x_1)) = 2 + x_1
   POL(i_1(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(h_1(x)) -> f_1(i_1(x))




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

(10)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(11) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(12)
YES