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:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), 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:

FOLDB(t, s(n)) → F(foldB(t, n), B)
FOLDB(t, s(n)) → FOLDB(t, n)
FOLDC(t, s(n)) → F(foldC(t, n), C)
FOLDC(t, s(n)) → FOLDC(t, n)
F(t, x) → F'(t, g(x))
F(t, x) → G(x)
F'(triple(a, b, c), B) → F(triple(a, b, c), A)
F'(triple(a, b, c), A) → F''(foldB(triple(s(a), 0, c), b))
F'(triple(a, b, c), A) → FOLDB(triple(s(a), 0, c), b)
F''(triple(a, b, c)) → FOLDC(triple(a, b, 0), c)
FOLD(t, x, s(n)) → F(fold(t, x, n), x)
FOLD(t, x, s(n)) → FOLD(t, x, n)

The TRS R consists of the following rules:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), 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 2 SCCs with 2 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

F(t, x) → F'(t, g(x))
F'(triple(a, b, c), B) → F(triple(a, b, c), A)
F'(triple(a, b, c), A) → F''(foldB(triple(s(a), 0, c), b))
F''(triple(a, b, c)) → FOLDC(triple(a, b, 0), c)
FOLDC(t, s(n)) → F(foldC(t, n), C)
FOLDC(t, s(n)) → FOLDC(t, n)
F'(triple(a, b, c), A) → FOLDB(triple(s(a), 0, c), b)
FOLDB(t, s(n)) → F(foldB(t, n), B)
FOLDB(t, s(n)) → FOLDB(t, n)

The TRS R consists of the following rules:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), x)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FOLDC(t, s(n)) → FOLDC(t, n)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x0, x1, x2)  =  F(x0, x1)
F'(x0, x1, x2)  =  F'(x0)
F''(x0, x1)  =  F''(x0)
FOLDC(x0, x1, x2)  =  FOLDC(x0)
FOLDB(x0, x1, x2)  =  FOLDB(x0)

Tags:
F has argument tags [10,0,8] and root tag 0
F' has argument tags [10,1,0] and root tag 0
F'' has argument tags [10,8] and root tag 0
FOLDC has argument tags [10,0,15] and root tag 0
FOLDB has argument tags [10,3,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A) = 0   
POL(B) = 0   
POL(C) = 1   
POL(F(x₁, x₂)) = 1 + x₁ + x₂   
POL(F'(x₁, x₂)) = 1 + x₁ + x₂   
POL(F''(x₁)) = 1 + x₁   
POL(FOLDB(x₁, x₂)) = 1 + x₁   
POL(FOLDC(x₁, x₂)) = 1 + x₁ + x₂   
POL(f(x₁, x₂)) = x₁ + x₂   
POL(f'(x₁, x₂)) = x₁ + x₂   
POL(f''(x₁)) = x₁   
POL(foldB(x₁, x₂)) = x₁   
POL(foldC(x₁, x₂)) = x₁ + x₂   
POL(g(x₁)) = x₁   
POL(s(x₁)) = 1 + x₁   
POL(triple(x₁, x₂, x₃)) = x₃   

The following usable rules [FROCOS05] were oriented:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
f'(triple(a, b, c), C) → triple(a, b, s(c))

(7) Obligation:

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

F(t, x) → F'(t, g(x))
F'(triple(a, b, c), B) → F(triple(a, b, c), A)
F'(triple(a, b, c), A) → F''(foldB(triple(s(a), 0, c), b))
F''(triple(a, b, c)) → FOLDC(triple(a, b, 0), c)
FOLDC(t, s(n)) → F(foldC(t, n), C)
F'(triple(a, b, c), A) → FOLDB(triple(s(a), 0, c), b)
FOLDB(t, s(n)) → F(foldB(t, n), B)
FOLDB(t, s(n)) → FOLDB(t, n)

The TRS R consists of the following rules:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), x)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F'(triple(a, b, c), A) → F''(foldB(triple(s(a), 0, c), b))
F''(triple(a, b, c)) → FOLDC(triple(a, b, 0), c)
FOLDC(t, s(n)) → F(foldC(t, n), C)
F'(triple(a, b, c), A) → FOLDB(triple(s(a), 0, c), b)
FOLDB(t, s(n)) → F(foldB(t, n), B)
FOLDB(t, s(n)) → FOLDB(t, n)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x0, x1, x2)  =  F(x0)
F'(x0, x1, x2)  =  F'(x0)
F''(x0, x1)  =  F''(x0)
FOLDC(x0, x1, x2)  =  FOLDC(x0)
FOLDB(x0, x1, x2)  =  FOLDB(x0, x2)

Tags:
F has argument tags [9,8,14] and root tag 4
F' has argument tags [9,0,3] and root tag 4
F'' has argument tags [12,4] and root tag 0
FOLDC has argument tags [9,1,13] and root tag 6
FOLDB has argument tags [0,3,12] and root tag 6

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A) = 1   
POL(B) = 1   
POL(C) = 1   
POL(F(x₁, x₂)) = x₁ + x₂   
POL(F'(x₁, x₂)) = x₁ + x₂   
POL(F''(x₁)) = x₁   
POL(FOLDB(x₁, x₂)) = 1 + x₁ + x₂   
POL(FOLDC(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = x₁ + x₂   
POL(f'(x₁, x₂)) = x₁ + x₂   
POL(f''(x₁)) = x₁   
POL(foldB(x₁, x₂)) = x₁ + x₂   
POL(foldC(x₁, x₂)) = x₁ + x₂   
POL(g(x₁)) = x₁   
POL(s(x₁)) = 1 + x₁   
POL(triple(x₁, x₂, x₃)) = x₂ + x₃   

The following usable rules [FROCOS05] were oriented:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
f'(triple(a, b, c), C) → triple(a, b, s(c))

(9) Obligation:

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

F(t, x) → F'(t, g(x))
F'(triple(a, b, c), B) → F(triple(a, b, c), A)

The TRS R consists of the following rules:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), x)

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

F(t, x) → F'(t, g(x))
F'(triple(a, b, c), B) → F(triple(a, b, c), A)

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

Tags:
F has argument tags [2,0,1] and root tag 1
F' has argument tags [4,3,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A) = 0   
POL(B) = 1   
POL(C) = 1   
POL(F(x₁, x₂)) = x₂   
POL(F'(x₁, x₂)) = 1 + x₁   
POL(g(x₁)) = x₁   
POL(triple(x₁, x₂, x₃)) = 0   

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), x)

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

FOLD(t, x, s(n)) → FOLD(t, x, n)

The TRS R consists of the following rules:

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), x)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FOLD(t, x, s(n)) → FOLD(t, x, n)

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

Tags:
FOLD has argument tags [1,2,0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(FOLD(x₁, x₂, x₃)) = 1 + x₃   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

g(A) → A
g(B) → A
g(B) → B
g(C) → A
g(C) → B
g(C) → C
foldB(t, 0) → t
foldB(t, s(n)) → f(foldB(t, n), B)
foldC(t, 0) → t
foldC(t, s(n)) → f(foldC(t, n), C)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, s(c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) → foldC(triple(a, b, 0), c)
fold(t, x, 0) → t
fold(t, x, s(n)) → f(fold(t, x, n), x)

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE