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

(0) Obligation:

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

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

Tuples:

F(z0, g(z0)) → c
F(z0, h(z1)) → c1(F(h(z0), z1))

S tuples:

F(z0, g(z0)) → c
F(z0, h(z1)) → c1(F(h(z0), z1))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1

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

Removed 1 trailing nodes:

F(z0, g(z0)) → c

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(z0, h(z1)) → c1(F(h(z0), z1))

S tuples:

F(z0, h(z1)) → c1(F(h(z0), z1))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, h(z1)) → c1(F(h(z0), z1))

S tuples:

F(z0, h(z1)) → c1(F(h(z0), z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c1

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

F(z0, h(z1)) → c1(F(h(z0), z1))

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

F(z0, h(z1)) → c1(F(h(z0), z1))

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

POL(F(x₁, x₂)) = [2]x₂²   
POL(c1(x₁)) = x₁   
POL(h(x₁)) = [2] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, h(z1)) → c1(F(h(z0), z1))

S tuples:none
K tuples:

F(z0, h(z1)) → c1(F(h(z0), z1))

Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c1

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

The set S is empty

(10) BOUNDS(1, 1)