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(a, b)
b → c

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(a, b)
b → c

Tuples:

F(z0, z0) → c1(F(a, b), B)

S tuples:

F(z0, z0) → c1(F(a, b), B)

K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Use narrowing to replace F(z0, z0) → c1(F(a, b), B) by

F(x0, x0) → c1(F(a, c), B)
F(x0, x0) → c1

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(a, b)
b → c

Tuples:

F(x0, x0) → c1(F(a, c), B)
F(x0, x0) → c1

S tuples:

F(x0, x0) → c1(F(a, c), B)
F(x0, x0) → c1

K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F

Compound Symbols:

c1, c1

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

Removed 2 trailing nodes:

F(x0, x0) → c1
F(x0, x0) → c1(F(a, c), B)

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(a, b)
b → c

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

f, b

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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