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:

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
admit(x, .(u, .(v, .(w, z)))) →⁺ cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,1,1].
The pumping substitution is [z / .(u, .(v, .(w, z)))].
The result substitution is [x / carry(x, u, v)].

(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:

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(='(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
admit/0
='/0
='/1
sum/0
sum/1
sum/2
carry/0
carry/1
carry/2

(6) Obligation:

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

admit(nil) → nil
admit(.(u, .(v, .(w, z)))) → cond(=', .(u, .(v, .(w, admit(z)))))
cond(true, y) → y

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
admit(nil) → nil
admit(.(u, .(v, .(w, z)))) → cond(=', .(u, .(v, .(w, admit(z)))))
cond(true, y) → y

Types:
admit :: nil:. → nil:.
nil :: nil:.
. :: w → nil:. → nil:.
w :: w
cond :: =':true → nil:. → nil:.
=' :: =':true
true :: =':true
hole_nil:.1_0 :: nil:.
hole_w2_0 :: w
hole_=':true3_0 :: =':true
gen_nil:.4_0 :: Nat → nil:.

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

Heuristically decided to analyse the following defined symbols:
admit

(10) Obligation:

TRS:
Rules:
admit(nil) → nil
admit(.(u, .(v, .(w, z)))) → cond(=', .(u, .(v, .(w, admit(z)))))
cond(true, y) → y

Types:
admit :: nil:. → nil:.
nil :: nil:.
. :: w → nil:. → nil:.
w :: w
cond :: =':true → nil:. → nil:.
=' :: =':true
true :: =':true
hole_nil:.1_0 :: nil:.
hole_w2_0 :: w
hole_=':true3_0 :: =':true
gen_nil:.4_0 :: Nat → nil:.

Generator Equations:
gen_nil:.4_0(0) ⇔ nil
gen_nil:.4_0(+(x, 1)) ⇔ .(w, gen_nil:.4_0(x))

The following defined symbols remain to be analysed:
admit

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

Proved the following rewrite lemma:
admit(gen_nil:.4_0(+(3, *(3, n6_0)))) → *5_0, rt ∈ Ω(n6₀)

Induction Base:
admit(gen_nil:.4_0(+(3, *(3, 0))))

Induction Step:
admit(gen_nil:.4_0(+(3, *(3, +(n6_0, 1))))) →_R^Ω(1)
cond(=', .(w, .(w, .(w, admit(gen_nil:.4_0(+(3, *(3, n6_0)))))))) →_IH
cond(=', .(w, .(w, .(w, *5_0))))

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
admit(nil) → nil
admit(.(u, .(v, .(w, z)))) → cond(=', .(u, .(v, .(w, admit(z)))))
cond(true, y) → y

Types:
admit :: nil:. → nil:.
nil :: nil:.
. :: w → nil:. → nil:.
w :: w
cond :: =':true → nil:. → nil:.
=' :: =':true
true :: =':true
hole_nil:.1_0 :: nil:.
hole_w2_0 :: w
hole_=':true3_0 :: =':true
gen_nil:.4_0 :: Nat → nil:.

Lemmas:
admit(gen_nil:.4_0(+(3, *(3, n6_0)))) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_nil:.4_0(0) ⇔ nil
gen_nil:.4_0(+(x, 1)) ⇔ .(w, gen_nil:.4_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
admit(gen_nil:.4_0(+(3, *(3, n6_0)))) → *5_0, rt ∈ Ω(n6₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
admit(nil) → nil
admit(.(u, .(v, .(w, z)))) → cond(=', .(u, .(v, .(w, admit(z)))))
cond(true, y) → y

Types:
admit :: nil:. → nil:.
nil :: nil:.
. :: w → nil:. → nil:.
w :: w
cond :: =':true → nil:. → nil:.
=' :: =':true
true :: =':true
hole_nil:.1_0 :: nil:.
hole_w2_0 :: w
hole_=':true3_0 :: =':true
gen_nil:.4_0 :: Nat → nil:.

Lemmas:
admit(gen_nil:.4_0(+(3, *(3, n6_0)))) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_nil:.4_0(0) ⇔ nil
gen_nil:.4_0(+(x, 1)) ⇔ .(w, gen_nil:.4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
admit(gen_nil:.4_0(+(3, *(3, n6_0)))) → *5_0, rt ∈ Ω(n6₀)

(18) BOUNDS(n^1, INF)