The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))

S tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))

K tuples:none
Defined Rule Symbols:

f, h, g

Defined Pair Symbols:

F, H, G

Compound Symbols:

c, c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

H(g(z0)) → c1(H(a))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

S tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

K tuples:none
Defined Rule Symbols:

f, h, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, 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.

F(a, z0) → c(F(g(z0), z0), G(z0))

We considered the (Usable) Rules:

g(h(z0)) → g(z0)

And the Tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

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

POL(F(x₁, x₂)) = [4]x₁   
POL(G(x₁)) = [3]   
POL(a) = [4]   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁)) = x₁   
POL(g(x₁)) = 0   
POL(h(x₁)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

S tuples:

G(h(z0)) → c3(G(z0))

K tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))

Defined Rule Symbols:

f, h, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c3

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

G(h(z0)) → c3(G(z0))

We considered the (Usable) Rules:

g(h(z0)) → g(z0)

And the Tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

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

POL(F(x₁, x₂)) = x₁·x₂   
POL(G(x₁)) = x₁   
POL(a) = [2]   
POL(c(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁)) = x₁   
POL(g(x₁)) = [1]   
POL(h(x₁)) = [2] + [2]x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

S tuples:none
K tuples:

F(a, z0) → c(F(g(z0), z0), G(z0))
G(h(z0)) → c3(G(z0))

Defined Rule Symbols:

f, h, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c3

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))