YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 64 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 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)) = x_1
   POL(false) = 1
   POL(if(x_1, x_2, x_3)) = x_1 + x_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




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

(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

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

(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

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

(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:
Polynomial interpretation [POLO]:

   POL(if(x_1, x_2, x_3)) = 2 + x_1 + x_2
   POL(true) = 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




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

(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