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

(0) Obligation:

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

f(x, x) → f(g(x), x)
g(x) → s(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)

Tuples:

F(z0, z0) → c(F(g(z0), z0), G(z0))

S tuples:

F(z0, z0) → c(F(g(z0), z0), G(z0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)

Tuples:

F(z0, z0) → c(F(g(z0), z0))

S tuples:

F(z0, z0) → c(F(g(z0), z0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(z0, z0) → c(F(g(z0), z0)) by

F(z0, z0) → c(F(s(z0), z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)

Tuples:

F(z0, z0) → c(F(s(z0), z0))

S tuples:

F(z0, z0) → c(F(s(z0), z0))

K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

(7) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

F(z0, z0) → c(F(s(z0), z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:none

Compound Symbols:none

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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