(0) Obligation:
Runtime Complexity TRS:
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)
Rewrite Strategy: FULL
(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)
Transformed TRS to relative TRS where S is empty.
(2) Obligation:
Runtime Complexity Relative TRS:
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)
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
triple/0
(4) Obligation:
Runtime Complexity Relative TRS:
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(b, c), C) → triple(b, cons(C, c))
f'(triple(b, c), B) → f(triple(b, c), A)
f'(triple(b, c), A) → f''(foldf(triple(nil, c), b))
f''(triple(b, c)) → foldf(triple(b, nil), c)
S is empty.
Rewrite Strategy: FULL
(5) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
foldf(x, cons(y, z)) →+ f(foldf(x, z), y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [z / cons(y, z)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)