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(x, y) → g(x, y)
g(h(x), y) → h(f(x, y))
g(h(x), y) → h(g(x, y))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z1) → g(z0, z1)
g(h(z0), z1) → h(f(z0, z1))
g(h(z0), z1) → h(g(z0, z1))

Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

S tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

(3) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(z0, z1) → g(z0, z1)
g(h(z0), z1) → h(f(z0, z1))
g(h(z0), z1) → h(g(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

S tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

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

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

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

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

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

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

POL(F(x₁, x₂)) = [1] + [2]x₁ + x₂   
POL(G(x₁, x₂)) = [2]x₁ + x₂   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(h(x₁)) = [1] + x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

S tuples:none
K tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

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