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:

:(:(x, y), z) → :(x, :(y, z))
:(+(x, y), z) → +(:(x, z), :(y, z))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
:(+(x, y), z) →⁺ +(:(x, z), :(y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / +(x, y)].
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:

:(:(x, y), z) → :(x, :(y, z))
:(+'(x, y), z) → +'(:(x, z), :(y, z))
:(z, +'(x, f(y))) → :(g(z, y), +'(x, a))

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
f/0
g/0
g/1

(6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

:(:(x, y), z) → :(x, :(y, z))
:(+'(x, y), z) → +'(:(x, z), :(y, z))
:(z, +'(x, f)) → :(g, +'(x, a))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
:(:(x, y), z) → :(x, :(y, z))
:(+'(x, y), z) → +'(:(x, z), :(y, z))
:(z, +'(x, f)) → :(g, +'(x, a))

Types:
: :: +':f:g:a → +':f:g:a → +':f:g:a
+' :: +':f:g:a → +':f:g:a → +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat → +':f:g:a

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
:

(10) Obligation:

TRS:
Rules:
:(:(x, y), z) → :(x, :(y, z))
:(+'(x, y), z) → +'(:(x, z), :(y, z))
:(z, +'(x, f)) → :(g, +'(x, a))

Types:
: :: +':f:g:a → +':f:g:a → +':f:g:a
+' :: +':f:g:a → +':f:g:a → +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat → +':f:g:a

Generator Equations:
gen_+':f:g:a2_0(0) ⇔ f
gen_+':f:g:a2_0(+(x, 1)) ⇔ +'(gen_+':f:g:a2_0(x), f)

The following defined symbols remain to be analysed:
:

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
:(gen_+':f:g:a2_0(+(1, 0)), gen_+':f:g:a2_0(b))

Induction Step:
:(gen_+':f:g:a2_0(+(1, +(n4_0, 1))), gen_+':f:g:a2_0(b)) →_R^Ω(1)
+'(:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)), :(f, gen_+':f:g:a2_0(b))) →_IH
+'(*3_0, :(f, gen_+':f:g:a2_0(b)))

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
:(:(x, y), z) → :(x, :(y, z))
:(+'(x, y), z) → +'(:(x, z), :(y, z))
:(z, +'(x, f)) → :(g, +'(x, a))

Types:
: :: +':f:g:a → +':f:g:a → +':f:g:a
+' :: +':f:g:a → +':f:g:a → +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat → +':f:g:a

Lemmas:
:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_+':f:g:a2_0(0) ⇔ f
gen_+':f:g:a2_0(+(x, 1)) ⇔ +'(gen_+':f:g:a2_0(x), f)

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)) → *3_0, rt ∈ Ω(n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
:(:(x, y), z) → :(x, :(y, z))
:(+'(x, y), z) → +'(:(x, z), :(y, z))
:(z, +'(x, f)) → :(g, +'(x, a))

Types:
: :: +':f:g:a → +':f:g:a → +':f:g:a
+' :: +':f:g:a → +':f:g:a → +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat → +':f:g:a

Lemmas:
:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_+':f:g:a2_0(0) ⇔ f
gen_+':f:g:a2_0(+(x, 1)) ⇔ +'(gen_+':f:g:a2_0(x), f)

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)) → *3_0, rt ∈ Ω(n4₀)

(18) BOUNDS(n^1, INF)