/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 49 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 30 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) TRUE


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

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.

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

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)

The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)

The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( FIRST_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( activate_1(x_1) ) = x_1
POL( n__first_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( first_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( n__from_1(x_1) ) = x_1 + 2
POL( from_1(x_1) ) = x_1 + 2
POL( n__s_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = x_1 + 1
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( nil ) = 1
POL( ACTIVATE_1(x_1) ) = x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   first(0, X) -> nil
   first(X1, X2) -> n__first(X1, X2)
   s(X) -> n__s(X)
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))


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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVATE(x1)  =  x1

n__first(x1, x2)  =  n__first(x2)

FIRST(x1, x2)  =  x2

activate(x1)  =  activate(x1)

cons(x1, x2)  =  x2

first(x1, x2)  =  first(x2)

n__from(x1)  =  n__from

from(x1)  =  from

n__s(x1)  =  n__s

s(x1)  =  s

nil  =  nil


Knuth-Bendix order [KBO] with precedence:activate_1 > s > n__s
activate_1 > first_1 > n__first_1
activate_1 > from > n__from
activate_1 > nil

and weight map:

   s=1
   n__first_1=2
   n__from=2
   first_1=2
   from=2
   activate_1=0
   n__s=1
   nil=2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   first(0, X) -> nil
   first(X1, X2) -> n__first(X1, X2)
   s(X) -> n__s(X)


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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(n__s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(10)
TRUE