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:

a(lambda(x), y) → lambda(a(x, p(1, a(y, t))))
a(p(x, y), z) → p(a(x, z), a(y, z))
a(a(x, y), z) → a(x, a(y, z))
a(id, x) → x
a(1, id) → 1
a(t, id) → t
a(1, p(x, y)) → x
a(t, p(x, y)) → y

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:

a(lambda(x), y) → lambda(a(x, p(1, a(y, t))))
a(p(x, y), z) → p(a(x, z), a(y, z))
a(a(x, y), z) → a(x, a(y, z))
a(id, x) → x
a(1, id) → 1
a(t, id) → t
a(1, p(x, y)) → x
a(t, p(x, y)) → y

S is empty.
Rewrite Strategy: FULL

(3) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a(lambda(x), y) →⁺ lambda(a(x, p(1, a(y, t))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / lambda(x)].
The result substitution is [y / p(1, a(y, t))].

(4) BOUNDS(n^1, INF)