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:

a__c → a__f(g(c))
a__f(g(X)) → g(X)
mark(c) → a__c
mark(f(X)) → a__f(X)
mark(g(X)) → g(X)
a__c → c
a__f(X) → f(X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__c → a__f(g(c))
a__c → c
a__f(g(z0)) → g(z0)
a__f(z0) → f(z0)
mark(c) → a__c
mark(f(z0)) → a__f(z0)
mark(g(z0)) → g(z0)

Tuples:

A__C → c1(A__F(g(c)))
MARK(c) → c5(A__C)
MARK(f(z0)) → c6(A__F(z0))

S tuples:

A__C → c1(A__F(g(c)))
MARK(c) → c5(A__C)
MARK(f(z0)) → c6(A__F(z0))

K tuples:none
Defined Rule Symbols:

a__c, a__f, mark

Defined Pair Symbols:

A__C, MARK

Compound Symbols:

c1, c5, c6

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 3 of 3 dangling nodes:

A__C → c1(A__F(g(c)))
MARK(c) → c5(A__C)
MARK(f(z0)) → c6(A__F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__c → a__f(g(c))
a__c → c
a__f(g(z0)) → g(z0)
a__f(z0) → f(z0)
mark(c) → a__c
mark(f(z0)) → a__f(z0)
mark(g(z0)) → g(z0)

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

a__c, a__f, mark

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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