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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
  ↳13 RisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 RisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 RisEmptyProof (⇔)
    ↳18 TRUE

(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
foldf(x, nil) → x
foldf(x, cons(y, z)) → f(foldf(x, z), y)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, cons(C, c))
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldf(triple(cons(A, a), nil, c), b))
f''(triple(a, b, c)) → foldf(triple(a, b, nil), c)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A) = 0   
POL(B) = 1   
POL(C) = 2   
POL(cons(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = x1 + x2   
POL(f'(x1, x2)) = x1 + x2   
POL(f''(x1)) = x1   
POL(foldf(x1, x2)) = x1 + x2   
POL(g(x1)) = x1   
POL(nil) = 0   
POL(triple(x1, x2, x3)) = x1 + x2 + x3   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

g(B) → A
g(C) → A
g(C) → B
f'(triple(a, b, c), B) → f(triple(a, b, c), A)


(2) Obligation:

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

g(A) → A
g(B) → B
g(C) → C
foldf(x, nil) → x
foldf(x, cons(y, z)) → f(foldf(x, z), y)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, cons(C, c))
f'(triple(a, b, c), A) → f''(foldf(triple(cons(A, a), nil, c), b))
f''(triple(a, b, c)) → foldf(triple(a, b, nil), c)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A) = 2   
POL(B) = 0   
POL(C) = 0   
POL(cons(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = x1 + 2·x2   
POL(f'(x1, x2)) = x1 + 2·x2   
POL(f''(x1)) = 1 + x1   
POL(foldf(x1, x2)) = x1 + 2·x2   
POL(g(x1)) = x1   
POL(nil) = 0   
POL(triple(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f'(triple(a, b, c), A) → f''(foldf(triple(cons(A, a), nil, c), b))
f''(triple(a, b, c)) → foldf(triple(a, b, nil), c)


(4) Obligation:

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

g(A) → A
g(B) → B
g(C) → C
foldf(x, nil) → x
foldf(x, cons(y, z)) → f(foldf(x, z), y)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, cons(C, c))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A) = 0   
POL(B) = 0   
POL(C) = 0   
POL(cons(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = x1 + x2   
POL(f'(x1, x2)) = x1 + x2   
POL(foldf(x1, x2)) = 1 + x1 + x2   
POL(g(x1)) = x1   
POL(nil) = 0   
POL(triple(x1, x2, x3)) = x1 + x2 + x3   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

foldf(x, nil) → x


(6) Obligation:

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

g(A) → A
g(B) → B
g(C) → C
foldf(x, cons(y, z)) → f(foldf(x, z), y)
f(t, x) → f'(t, g(x))
f'(triple(a, b, c), C) → triple(a, b, cons(C, c))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A) = 2   
POL(B) = 0   
POL(C) = 2   
POL(cons(x1, x2)) = 1 + x1 + x2   
POL(f(x1, x2)) = x1 + 2·x2   
POL(f'(x1, x2)) = x1 + 2·x2   
POL(foldf(x1, x2)) = 2 + x1 + 2·x2   
POL(g(x1)) = x1   
POL(triple(x1, x2, x3)) = x1 + 2·x2 + x3   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

foldf(x, cons(y, z)) → f(foldf(x, z), y)
f'(triple(a, b, c), C) → triple(a, b, cons(C, c))


(8) Obligation:

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

g(A) → A
g(B) → B
g(C) → C
f(t, x) → f'(t, g(x))

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A) = 2   
POL(B) = 0   
POL(C) = 0   
POL(f(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(f'(x1, x2)) = 1 + x1 + x2   
POL(g(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

f(t, x) → f'(t, g(x))


(10) Obligation:

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

g(A) → A
g(B) → B
g(C) → C

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(A) = 0   
POL(B) = 0   
POL(C) = 0   
POL(g(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

g(A) → A
g(B) → B
g(C) → C


(12) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(13) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(14) TRUE

(15) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(16) TRUE

(17) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(18) TRUE