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

(0) Obligation:

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

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(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:

A__AND(true, X) → MARK(X)
A__IF(true, X, Y) → MARK(X)
A__IF(false, X, Y) → MARK(Y)
A__ADD(0, X) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → A__IF(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(add(X1, X2)) → A__ADD(mark(X1), X2)
MARK(add(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(from(X)) → A__FROM(X)

The TRS R consists of the following rules:

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(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 2 less nodes.

(4) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(true, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → A__IF(mark(X1), X2, X3)
A__IF(true, X, Y) → MARK(X)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(add(X1, X2)) → A__ADD(mark(X1), X2)
A__ADD(0, X) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
A__IF(false, X, Y) → MARK(Y)

The TRS R consists of the following rules:

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(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.

MARK(add(X1, X2)) → A__ADD(mark(X1), X2)
A__ADD(0, X) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0, x1)
A__AND(x0, x1, x2)  =  A__AND(x0, x1, x2)
A__IF(x0, x1, x2, x3)  =  A__IF(x0, x2, x3)
A__ADD(x0, x1, x2)  =  A__ADD(x0, x1, x2)

Tags:
MARK has argument tags [2,0] and root tag 0
A__AND has argument tags [2,13,0] and root tag 0
A__IF has argument tags [2,1,0,0] and root tag 0
A__ADD has argument tags [1,2,0] and root tag 3

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK
and(x1, x2)  =  and(x1, x2)
A__AND(x1, x2)  =  A__AND
mark(x1)  =  x1
true  =  true
if(x1, x2, x3)  =  if(x1, x2, x3)
A__IF(x1, x2, x3)  =  A__IF
add(x1, x2)  =  add(x1, x2)
A__ADD(x1, x2)  =  A__ADD
0  =  0
first(x1, x2)  =  first(x1, x2)
false  =  false
a__and(x1, x2)  =  a__and(x1, x2)
a__if(x1, x2, x3)  =  a__if(x1, x2, x3)
a__add(x1, x2)  =  a__add(x1, x2)
a__first(x1, x2)  =  a__first(x1, x2)
from(x1)  =  from(x1)
a__from(x1)  =  a__from(x1)
s(x1)  =  x1
nil  =  nil
cons(x1, x2)  =  x2

Lexicographic path order with status [LPO].
Quasi-Precedence:

[MARK, A_AND, A_IF, A_ADD, 0] > nil
[and₂, a_and2] > false
[if₃, a_if3]
[add₂, a_add2]
[first₂, a_first2]
[from₁, a_from1]

Status:

MARK: []
and₂: [2,1]
A_AND: []
true: []
if₃: [1,3,2]
A_IF: []
add₂: [2,1]
A_ADD: []
0: []
first₂: [1,2]
false: []
a_and2: [2,1]
a_if3: [1,3,2]
a_add2: [2,1]
a_first2: [1,2]
from₁: [1]
a_from1: [1]
nil: []

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(true, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → A__IF(mark(X1), X2, X3)
A__IF(true, X, Y) → MARK(X)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
A__IF(false, X, Y) → MARK(Y)

The TRS R consists of the following rules:

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(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.

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(true, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → A__IF(mark(X1), X2, X3)
A__IF(true, X, Y) → MARK(X)
MARK(if(X1, X2, X3)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
A__IF(false, X, Y) → MARK(Y)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x1)
A__AND(x0, x1, x2)  =  A__AND(x0, x1, x2)
A__IF(x0, x1, x2, x3)  =  A__IF(x0, x3)

Tags:
MARK has argument tags [1,4] and root tag 1
A__AND has argument tags [4,8,11] and root tag 0
A__IF has argument tags [14,0,8,0] and root tag 2

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
and(x1, x2)  =  and(x1, x2)
A__AND(x1, x2)  =  A__AND(x1, x2)
mark(x1)  =  x1
true  =  true
if(x1, x2, x3)  =  if(x1, x2, x3)
A__IF(x1, x2, x3)  =  A__IF(x1, x2, x3)
add(x1, x2)  =  add(x1, x2)
first(x1, x2)  =  first(x1, x2)
false  =  false
a__and(x1, x2)  =  a__and(x1, x2)
a__if(x1, x2, x3)  =  a__if(x1, x2, x3)
a__add(x1, x2)  =  a__add(x1, x2)
0  =  0
a__first(x1, x2)  =  a__first(x1, x2)
from(x1)  =  from(x1)
a__from(x1)  =  a__from(x1)
s(x1)  =  x1
nil  =  nil
cons(x1, x2)  =  x2

Lexicographic path order with status [LPO].
Quasi-Precedence:

true > [from₁, a_from1]
[if₃, a_if3] > A_IF3 > [from₁, a_from1]
[add₂, a_add2] > [MARK₁, first₂, a_first2] > [and₂, A_AND2, a_and2] > [from₁, a_from1]
[add₂, a_add2] > [MARK₁, first₂, a_first2] > A_IF3 > [from₁, a_from1]
[add₂, a_add2] > [MARK₁, first₂, a_first2] > nil > [from₁, a_from1]
false > [from₁, a_from1]
0 > nil > [from₁, a_from1]

Status:

MARK₁: [1]
and₂: [1,2]
A_AND2: [1,2]
true: []
if₃: [3,2,1]
A_IF3: [1,3,2]
add₂: [1,2]
first₂: [2,1]
false: []
a_and2: [1,2]
a_if3: [3,2,1]
a_add2: [1,2]
0: []
a_first2: [2,1]
from₁: [1]
a_from1: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(true, X) → mark(X)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
a__if(true, X, Y) → mark(X)
mark(add(X1, X2)) → a__add(mark(X1), X2)
a__add(0, X) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__first(X1, X2) → first(X1, X2)
a__and(false, Y) → false
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(s(X), Y) → s(add(X, Y))
a__add(X1, X2) → add(X1, X2)
a__from(X) → cons(X, from(s(X)))
a__from(X) → from(X)

(8) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__and(true, X) → mark(X)
a__and(false, Y) → false
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(add(X, Y))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
a__from(X) → cons(X, from(s(X)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(add(X1, X2)) → a__add(mark(X1), X2)
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(from(X)) → a__from(X)
mark(true) → true
mark(false) → false
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__add(X1, X2) → add(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)

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

(9) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(10) TRUE