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

(0) Obligation:

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

f(0, 1, g(x, y), z) → f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) → 0
g(0, 1) → 1
h(g(x, y)) → h(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, g(z0, z1), z2) → f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
g(0, 1) → 0
g(0, 1) → 1
h(g(z0, z1)) → h(z0)

Tuples:

F(0, 1, g(z0, z1), z2) → c(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
H(g(z0, z1)) → c3(H(z0))

S tuples:

F(0, 1, g(z0, z1), z2) → c(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
H(g(z0, z1)) → c3(H(z0))

K tuples:none
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F, H

Compound Symbols:

c, c3

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

F(0, 1, g(z0, z1), z2) → c(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, g(z0, z1), z2) → f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
g(0, 1) → 0
g(0, 1) → 1
h(g(z0, z1)) → h(z0)

Tuples:

H(g(z0, z1)) → c3(H(z0))

S tuples:

H(g(z0, z1)) → c3(H(z0))

K tuples:none
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

H

Compound Symbols:

c3

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

H(g(z0, z1)) → c3(H(z0))

We considered the (Usable) Rules:none
And the Tuples:

H(g(z0, z1)) → c3(H(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(H(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(g(x₁, x₂)) = [1] + [4]x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, g(z0, z1), z2) → f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
g(0, 1) → 0
g(0, 1) → 1
h(g(z0, z1)) → h(z0)

Tuples:

H(g(z0, z1)) → c3(H(z0))

S tuples:none
K tuples:

H(g(z0, z1)) → c3(H(z0))

Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

H

Compound Symbols:

c3

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))