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(f(X)) → c(f(g(f(X))))
c(X) → d(X)
h(X) → c(d(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(f(g(f(z0))))
c(z0) → d(z0)
h(z0) → c(d(z0))

Tuples:

F(f(z0)) → c1(C(f(g(f(z0)))), F(g(f(z0))), F(z0))
H(z0) → c3(C(d(z0)))

S tuples:

F(f(z0)) → c1(C(f(g(f(z0)))), F(g(f(z0))), F(z0))
H(z0) → c3(C(d(z0)))

K tuples:none
Defined Rule Symbols:

f, c, h

Defined Pair Symbols:

F, H

Compound Symbols:

c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0)) → c1(C(f(g(f(z0)))), F(g(f(z0))), F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(f(g(f(z0))))
c(z0) → d(z0)
h(z0) → c(d(z0))

Tuples:

H(z0) → c3(C(d(z0)))

S tuples:

H(z0) → c3(C(d(z0)))

K tuples:none
Defined Rule Symbols:

f, c, h

Defined Pair Symbols:

H

Compound Symbols:

c3

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

H(z0) → c3(C(d(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → c(f(g(f(z0))))
c(z0) → d(z0)
h(z0) → c(d(z0))

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

f, c, h

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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