The Runtime Complexity (innermost) 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: INNERMOST

(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: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
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

Types:
admit :: carry → nil:. → nil:.
nil :: nil:.
. :: w → nil:. → nil:.
w :: w
cond :: =':true → nil:. → nil:.
=' :: sum → w → =':true
sum :: carry → w → w → sum
carry :: carry → w → w → carry
true :: =':true
hole_nil:.1_0 :: nil:.
hole_carry2_0 :: carry
hole_w3_0 :: w
hole_=':true4_0 :: =':true
hole_sum5_0 :: sum
gen_nil:.6_0 :: Nat → nil:.
gen_carry7_0 :: Nat → carry

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

Heuristically decided to analyse the following defined symbols:
admit

(8) Obligation:

Innermost TRS:
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

Types:
admit :: carry → nil:. → nil:.
nil :: nil:.
. :: w → nil:. → nil:.
w :: w
cond :: =':true → nil:. → nil:.
=' :: sum → w → =':true
sum :: carry → w → w → sum
carry :: carry → w → w → carry
true :: =':true
hole_nil:.1_0 :: nil:.
hole_carry2_0 :: carry
hole_w3_0 :: w
hole_=':true4_0 :: =':true
hole_sum5_0 :: sum
gen_nil:.6_0 :: Nat → nil:.
gen_carry7_0 :: Nat → carry

Generator Equations:
gen_nil:.6_0(0) ⇔ nil
gen_nil:.6_0(+(x, 1)) ⇔ .(w, gen_nil:.6_0(x))
gen_carry7_0(0) ⇔ hole_carry2_0
gen_carry7_0(+(x, 1)) ⇔ carry(gen_carry7_0(x), w, w)

The following defined symbols remain to be analysed:
admit