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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 QDPOrderProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 QDPOrderProof (⇔)
↳14 QDP
↳15 DependencyGraphProof (⇔)
↳16 QDP
↳17 QDPOrderProof (⇔)
↳18 QDP
↳19 DependencyGraphProof (⇔)
↳20 QDP
↳21 UsableRulesProof (⇔)
↳22 QDP
↳23 QDPSizeChangeProof (⇔)
↳24 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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:

AND(true, X) → ACTIVATE(X)
IF(true, X, Y) → ACTIVATE(X)
IF(false, X, Y) → ACTIVATE(Y)
ADD(0, X) → ACTIVATE(X)
ADD(s(X), Y) → S(n__add(activate(X), activate(Y)))
ADD(s(X), Y) → ACTIVATE(X)
ADD(s(X), Y) → ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
FROM(X) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
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(X)
ACTIVATE(n__s(X)) → S(X)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
ADD(0, X) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__from(X)) → FROM(X)
FROM(X) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)
ADD(s(X), Y) → ACTIVATE(Y)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__add(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(ADD(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(activate(x1)) = -I + 0A·x1

POL(0) = 0A

POL(n__first(x1, x2)) = 0A + 1A·x1 + 0A·x2

POL(FIRST(x1, x2)) = -I + 1A·x1 + 0A·x2

POL(s(x1)) = 0A + 0A·x1

POL(cons(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(n__from(x1)) = 0A + 0A·x1

POL(FROM(x1)) = 0A + 0A·x1

POL(add(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(first(x1, x2)) = 0A + 1A·x1 + 0A·x2

POL(nil) = 1A

POL(from(x1)) = 0A + 0A·x1

POL(n__s(x1)) = 0A + 0A·x1

The following usable rules [FROCOS05] were oriented:

add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
first(0, X) → nil
add(X1, X2) → n__add(X1, X2)
from(X) → cons(activate(X), n__from(n__s(activate(X))))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__add(X1, X2)) → add(activate(X1), X2)
add(0, X) → activate(X)
s(X) → n__s(X)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
activate(n__s(X)) → s(X)

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
ADD(0, X) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__from(X)) → FROM(X)
FROM(X) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)
ADD(s(X), Y) → ACTIVATE(Y)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__from(X)) → FROM(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = -I + 0A·x1

POL(n__add(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(ADD(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(activate(x1)) = -I + 0A·x1

POL(0) = 0A

POL(n__first(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(FIRST(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(s(x1)) = -I + 0A·x1

POL(cons(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(n__from(x1)) = -I + 1A·x1

POL(FROM(x1)) = -I + 0A·x1

POL(add(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(first(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(nil) = 0A

POL(from(x1)) = -I + 1A·x1

POL(n__s(x1)) = -I + 0A·x1

The following usable rules [FROCOS05] were oriented:

add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
first(0, X) → nil
add(X1, X2) → n__add(X1, X2)
from(X) → cons(activate(X), n__from(n__s(activate(X))))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__add(X1, X2)) → add(activate(X1), X2)
add(0, X) → activate(X)
s(X) → n__s(X)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
activate(n__s(X)) → s(X)

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
ADD(0, X) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FROM(X) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)
ADD(s(X), Y) → ACTIVATE(Y)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD(0, X) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(Y)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ADD(s(X), Y) → ACTIVATE(Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ADD(x1, x2)) = 0A + 0A·x1 + 1A·x2

POL(0) = 0A

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__add(x1, x2)) = -I + 0A·x1 + 1A·x2

POL(activate(x1)) = 0A + 0A·x1

POL(s(x1)) = 1A + 0A·x1

POL(n__first(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(FIRST(x1, x2)) = -I + -I·x1 + 0A·x2

POL(cons(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(add(x1, x2)) = -I + 0A·x1 + 1A·x2

POL(first(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(nil) = 0A

POL(from(x1)) = 1A + 0A·x1

POL(n__from(x1)) = 1A + 0A·x1

POL(n__s(x1)) = 1A + 0A·x1

The following usable rules [FROCOS05] were oriented:

add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
first(0, X) → nil
add(X1, X2) → n__add(X1, X2)
from(X) → cons(activate(X), n__from(n__s(activate(X))))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__add(X1, X2)) → add(activate(X1), X2)
add(0, X) → activate(X)
s(X) → n__s(X)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
activate(n__s(X)) → s(X)

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD(0, X) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ADD(x1, x2)) = 0A + 4A·x1 + 0A·x2

POL(0) = 0A

POL(ACTIVATE(x1)) = 4A + 0A·x1

POL(n__add(x1, x2)) = 0A + 5A·x1 + 3A·x2

POL(activate(x1)) = -I + 0A·x1

POL(s(x1)) = 0A + 0A·x1

POL(n__first(x1, x2)) = 0A + 0A·x1 + 1A·x2

POL(FIRST(x1, x2)) = 4A + 0A·x1 + 0A·x2

POL(cons(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(add(x1, x2)) = 0A + 5A·x1 + 3A·x2

POL(first(x1, x2)) = 0A + 0A·x1 + 1A·x2

POL(nil) = 0A

POL(from(x1)) = 0A + 0A·x1

POL(n__from(x1)) = 0A + 0A·x1

POL(n__s(x1)) = 0A + 0A·x1

The following usable rules [FROCOS05] were oriented:

add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
first(0, X) → nil
add(X1, X2) → n__add(X1, X2)
from(X) → cons(activate(X), n__from(n__s(activate(X))))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__add(X1, X2)) → add(activate(X1), X2)
add(0, X) → activate(X)
s(X) → n__s(X)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
activate(n__s(X)) → s(X)

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD(0, X) → ACTIVATE(X)
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(16) 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(Y)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FIRST(s(X), cons(Y, Z)) → ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = -I + 1A·x1

POL(n__first(x1, x2)) = -I + 0A·x1 + 2A·x2

POL(FIRST(x1, x2)) = -I + 1A·x1 + 2A·x2

POL(activate(x1)) = -I + 0A·x1

POL(s(x1)) = -I + 0A·x1

POL(cons(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(n__add(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(add(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(first(x1, x2)) = -I + 0A·x1 + 2A·x2

POL(0) = 5A

POL(nil) = 0A

POL(from(x1)) = -I + 0A·x1

POL(n__from(x1)) = -I + 0A·x1

POL(n__s(x1)) = -I + 0A·x1

The following usable rules [FROCOS05] were oriented:

add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
first(0, X) → nil
add(X1, X2) → n__add(X1, X2)
from(X) → cons(activate(X), n__from(n__s(activate(X))))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__add(X1, X2)) → add(activate(X1), X2)
add(0, X) → activate(X)
s(X) → n__s(X)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
activate(n__s(X)) → s(X)

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

and(true, X) → activate(X)
and(false, Y) → false
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
add(0, X) → activate(X)
add(s(X), Y) → s(n__add(activate(X), activate(Y)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(activate(Y), n__first(activate(X), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
add(X1, X2) → n__add(X1, X2)
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__add(X1, X2)) → add(activate(X1), X2)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(X)
activate(n__s(X)) → s(X)
activate(X) → X

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

(21) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)

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

(23) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
    The graph contains the following edges 1 > 1

  • ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
    The graph contains the following edges 1 > 1

(24) TRUE