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:

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1) → z0
g(a) → h(a, b, a)
i(z0) → f(z0, z0)
h(z0, z0, z1) → g(z0)

Tuples:

G(a) → c1(H(a, b, a))
I(z0) → c2(F(z0, z0))
H(z0, z0, z1) → c3(G(z0))

S tuples:

G(a) → c1(H(a, b, a))
I(z0) → c2(F(z0, z0))
H(z0, z0, z1) → c3(G(z0))

K tuples:none
Defined Rule Symbols:

f, g, i, h

Defined Pair Symbols:

G, I, H

Compound Symbols:

c1, c2, c3

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

H(z0, z0, z1) → c3(G(z0))
G(a) → c1(H(a, b, a))
I(z0) → c2(F(z0, z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1) → z0
g(a) → h(a, b, a)
i(z0) → f(z0, z0)
h(z0, z0, z1) → g(z0)

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

f, g, i, h

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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