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(a, b) → f(a, c)
f(c, d) → f(b, d)

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(a, c)
f(c, d) → f(b, d)

Tuples:

F(a, b) → c1(F(a, c))
F(c, d) → c2(F(b, d))

S tuples:

F(a, b) → c1(F(a, c))
F(c, d) → c2(F(b, d))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

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

Removed 2 trailing nodes:

F(c, d) → c2(F(b, d))
F(a, b) → c1(F(a, c))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a, b) → f(a, c)
f(c, d) → f(b, d)

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

f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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