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(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → b
f(c) → d
f(g(z0, z1)) → g(f(z0), f(z1))
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1)))
g(z0, z0) → h(e, z0)

Tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1))
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))

S tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1))
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

(3) 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.

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1))
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))

We considered the (Usable) Rules:

f(a) → b
f(c) → d
f(g(z0, z1)) → g(f(z0), f(z1))
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1)))
g(z0, z0) → h(e, z0)

And the Tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1))
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))

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

POL(F(x₁)) = [1] + [2]x₁   
POL(G(x₁, x₂)) = 0   
POL(a) = [4]   
POL(b) = [3]   
POL(c) = [3]   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(d) = [5]   
POL(e) = [3]   
POL(f(x₁)) = [1]   
POL(g(x₁, x₂)) = [4] + [4]x₁ + [4]x₂   
POL(h(x₁, x₂)) = [4] + x₁ + x₂   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → b
f(c) → d
f(g(z0, z1)) → g(f(z0), f(z1))
f(h(z0, z1)) → g(h(z1, f(z0)), h(z0, f(z1)))
g(z0, z0) → h(e, z0)

Tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1))
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))

S tuples:none
K tuples:

F(g(z0, z1)) → c3(G(f(z0), f(z1)), F(z0), F(z1))
F(h(z0, z1)) → c4(G(h(z1, f(z0)), h(z0, f(z1))), F(z0), F(z1))

Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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