The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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) 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 [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(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(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

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
foldf, f, f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(8) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
f, foldf, f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(10) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
f', foldf, f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f'.

(12) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
f'', foldf

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f''.

(14) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
foldf

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

Induction Base:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(0)) →_R^Ω(1)
triple(nil, nil, nil)

Induction Step:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(+(n182_0, 1))) →_R^Ω(1)
f(foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)), A) →_IH
f(triple(gen_nil:cons4_0(c183_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), A) →_R^Ω(1)
f'(triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), g(A)) →_R^Ω(1)
f'(triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), A) →_R^Ω(1)
f''(foldf(triple(cons(A, gen_nil:cons4_0(n182_0)), nil, gen_nil:cons4_0(0)), gen_nil:cons4_0(0))) →_R^Ω(1)
f''(triple(cons(A, gen_nil:cons4_0(n182_0)), nil, gen_nil:cons4_0(0))) →_R^Ω(1)
foldf(triple(cons(A, gen_nil:cons4_0(n182_0)), nil, nil), gen_nil:cons4_0(0)) →_R^Ω(1)
triple(cons(A, gen_nil:cons4_0(n182_0)), nil, nil)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
f, f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(19) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
f', f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f'.

(21) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
f''

They will be analysed ascendingly in the following order:
foldf = f
foldf = f'
foldf = f''
f = f'
f = f''
f' = f''

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f''.

(23) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
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)

Types:
g :: A:B:C → A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldf :: triple → nil:cons → triple
nil :: nil:cons
cons :: A:B:C → nil:cons → nil:cons
f :: triple → A:B:C → triple
f' :: triple → A:B:C → triple
triple :: nil:cons → nil:cons → nil:cons → triple
f'' :: triple → triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_nil:cons3_0 :: nil:cons
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(A, gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
foldf(triple(nil, nil, nil), gen_nil:cons4_0(n182_0)) → triple(gen_nil:cons4_0(n182_0), gen_nil:cons4_0(0), gen_nil:cons4_0(0)), rt ∈ Ω(1 + n182₀)

(28) BOUNDS(n^1, INF)