(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) = 2
POL(B) = 2
POL(C) = 2
POL(cons(x1, x2)) = x1 + x2
POL(f(x1, x2)) = x1 + 2·x2
POL(f'(x1, x2)) = x1 + 2·x2
POL(f''(x1)) = 2 + 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)) → foldf(triple(a, b, nil), c)
(2) 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))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(A) = 0
POL(B) = 0
POL(C) = 1
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(C) → A
g(C) → B
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(A) → A
g(B) → 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), B) → f(triple(a, b, c), A)
f'(triple(a, b, c), A) → f''(foldf(triple(cons(A, a), nil, c), b))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(A) = 0
POL(B) = 1
POL(C) = 0
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
f'(triple(a, b, c), B) → f(triple(a, b, c), A)
(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, 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))
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(A) = 1
POL(B) = 2
POL(C) = 0
POL(cons(x1, x2)) = 1 + x1 + x2
POL(f(x1, x2)) = 2 + x1 + 2·x2
POL(f'(x1, x2)) = 1 + x1 + 2·x2
POL(f''(x1)) = x1
POL(foldf(x1, x2)) = x1 + 2·x2
POL(g(x1)) = x1
POL(nil) = 0
POL(triple(x1, x2, x3)) = 1 + x1 + 2·x2 + x3
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))
f'(triple(a, b, c), A) → f''(foldf(triple(cons(A, a), nil, c), b))
(8) 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'(triple(a, b, c), C) → triple(a, b, cons(C, c))
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
g1 > f'2 > triple3 > A > B > foldf2 > f2 > nil > cons2 > C
and weight map:
A=1
B=1
C=1
nil=1
g_1=0
foldf_2=0
cons_2=0
f_2=0
triple_3=0
f'_2=0
The variable weight is 1With 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
foldf(x, nil) → x
foldf(x, cons(y, z)) → f(foldf(x, z), y)
f'(triple(a, b, c), C) → triple(a, b, cons(C, c))
(10) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(11) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(12) YES