/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 53 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 6 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   active(and(true, X)) -> mark(X)
   active(and(false, Y)) -> mark(false)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(add(X1, X2)) -> add(active(X1), X2)
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   add(mark(X1), X2) -> mark(add(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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

   active(and(true, X)) -> mark(X)
   active(and(false, Y)) -> mark(false)
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(add(X1, X2)) -> add(active(X1), X2)
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   add(mark(X1), X2) -> mark(add(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
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:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
0: empty set
s: empty set
first: {1, 2}
nil: empty set
cons: empty set
from: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
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(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
0: empty set
s: empty set
first: {1, 2}
nil: empty set
cons: empty set
from: 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:

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
0: empty set
s: empty set
first: {1, 2}
nil: empty set
cons: empty set
from: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))




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

   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

s: empty set
cons: empty set
from: 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:

   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

s: empty set
cons: empty set
from: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   from(X) -> cons(X, from(s(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