The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(lambda(z0), z1) → lambda(a(z0, 1))
a(lambda(z0), z1) → lambda(a(z0, a(z1, t)))
a(a(z0, z1), z2) → a(z0, a(z1, z2))
a(z0, z1) → z0
a(z0, z1) → z1
lambda(z0) → z0

Tuples:

A(lambda(z0), z1) → c(LAMBDA(a(z0, 1)), A(z0, 1))
A(lambda(z0), z1) → c1(LAMBDA(a(z0, a(z1, t))), A(z0, a(z1, t)), A(z1, t))
A(a(z0, z1), z2) → c2(A(z0, a(z1, z2)), A(z1, z2))

S tuples:

A(lambda(z0), z1) → c(LAMBDA(a(z0, 1)), A(z0, 1))
A(lambda(z0), z1) → c1(LAMBDA(a(z0, a(z1, t))), A(z0, a(z1, t)), A(z1, t))
A(a(z0, z1), z2) → c2(A(z0, a(z1, z2)), A(z1, z2))

K tuples:none
Defined Rule Symbols:

a, lambda

Defined Pair Symbols:

A

Compound Symbols:

c, c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

A(lambda(z0), z1) → c(LAMBDA(a(z0, 1)), A(z0, 1))
A(lambda(z0), z1) → c1(LAMBDA(a(z0, a(z1, t))), A(z0, a(z1, t)), A(z1, t))
A(a(z0, z1), z2) → c2(A(z0, a(z1, z2)), A(z1, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(lambda(z0), z1) → lambda(a(z0, 1))
a(lambda(z0), z1) → lambda(a(z0, a(z1, t)))
a(a(z0, z1), z2) → a(z0, a(z1, z2))
a(z0, z1) → z0
a(z0, z1) → z1
lambda(z0) → z0

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, lambda

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))