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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(g(f(a), h(a, f(a)))) → c(F(h(g(f(a), a), g(f(a), f(a)))), F(a), F(a), F(a))

S tuples:

F(g(f(a), h(a, f(a)))) → c(F(h(g(f(a), a), g(f(a), f(a)))), F(a), F(a), F(a))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(g(f(a), h(a, f(a)))) → c(F(h(g(f(a), a), g(f(a), f(a)))), F(a), F(a), F(a))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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))