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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSToCSRProof [SOUND, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 34 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(X)) -> mark(if(X, c, f(true)))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(f(X)) -> f(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
   f(mark(X)) -> mark(f(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
   proper(f(X)) -> f(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(c) -> ok(c)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   f(ok(X)) -> ok(f(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(f(x0))
   active(if(x0, x1, x2))
   f(mark(x0))
   if(mark(x0), x1, x2)
   if(x0, mark(x1), x2)
   proper(f(x0))
   proper(if(x0, x1, x2))
   proper(c)
   proper(true)
   proper(false)
   f(ok(x0))
   if(ok(x0), ok(x1), ok(x2))
   top(mark(x0))
   top(ok(x0))


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(1) QTRSToCSRProof (SOUND)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(X)) -> mark(if(X, c, f(true)))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(f(X)) -> f(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
   f(mark(X)) -> mark(f(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
   proper(f(X)) -> f(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(c) -> ok(c)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   f(ok(X)) -> ok(f(X))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(f(x0))
   active(if(x0, x1, x2))
   f(mark(x0))
   if(mark(x0), x1, x2)
   if(x0, mark(x1), x2)
   proper(f(x0))
   proper(if(x0, x1, x2))
   proper(c)
   proper(true)
   proper(false)
   f(ok(x0))
   if(ok(x0), ok(x1), ok(x2))
   top(mark(x0))
   top(ok(x0))

Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

f: {1}
if: {1, 2}
c: empty set
true: empty set
false: empty set
The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
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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
   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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(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
   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




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(4)
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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(5) 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))




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(6)
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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(7) 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)) = 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




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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