The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(X)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(z0)) → a__c(f(g(f(z0))))
a__f(z0) → f(z0)
a__c(z0) → d(z0)
a__c(z0) → c(z0)
a__h(z0) → a__c(d(z0))
a__h(z0) → h(z0)
mark(f(z0)) → a__f(mark(z0))
mark(c(z0)) → a__c(z0)
mark(h(z0)) → a__h(mark(z0))
mark(g(z0)) → g(z0)
mark(d(z0)) → d(z0)

Tuples:

A__F(f(z0)) → c1(A__C(f(g(f(z0)))))
A__H(z0) → c5(A__C(d(z0)))
MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(c(z0)) → c8(A__C(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

S tuples:

A__F(f(z0)) → c1(A__C(f(g(f(z0)))))
A__H(z0) → c5(A__C(d(z0)))
MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(c(z0)) → c8(A__C(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

K tuples:none
Defined Rule Symbols:

a__f, a__c, a__h, mark

Defined Pair Symbols:

A__F, A__H, MARK

Compound Symbols:

c1, c5, c7, c8, c9

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

Removed 3 trailing nodes:

A__F(f(z0)) → c1(A__C(f(g(f(z0)))))
MARK(c(z0)) → c8(A__C(z0))
A__H(z0) → c5(A__C(d(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(z0)) → a__c(f(g(f(z0))))
a__f(z0) → f(z0)
a__c(z0) → d(z0)
a__c(z0) → c(z0)
a__h(z0) → a__c(d(z0))
a__h(z0) → h(z0)
mark(f(z0)) → a__f(mark(z0))
mark(c(z0)) → a__c(z0)
mark(h(z0)) → a__h(mark(z0))
mark(g(z0)) → g(z0)
mark(d(z0)) → d(z0)

Tuples:

MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

S tuples:

MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

K tuples:none
Defined Rule Symbols:

a__f, a__c, a__h, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c7, c9

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

We considered the (Usable) Rules:

mark(f(z0)) → a__f(mark(z0))
mark(c(z0)) → a__c(z0)
mark(h(z0)) → a__h(mark(z0))
mark(g(z0)) → g(z0)
mark(d(z0)) → d(z0)
a__h(z0) → a__c(d(z0))
a__h(z0) → h(z0)
a__c(z0) → d(z0)
a__c(z0) → c(z0)
a__f(f(z0)) → a__c(f(g(f(z0))))
a__f(z0) → f(z0)

And the Tuples:

MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(A__F(x₁)) = 0   
POL(A__H(x₁)) = [2]   
POL(MARK(x₁)) = [2]x₁   
POL(a__c(x₁)) = [3] + [4]x₁   
POL(a__f(x₁)) = [5] + [2]x₁   
POL(a__h(x₁)) = [3] + [3]x₁   
POL(c(x₁)) = [3]   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(d(x₁)) = [3]   
POL(f(x₁)) = [3] + x₁   
POL(g(x₁)) = [3]   
POL(h(x₁)) = [4] + x₁   
POL(mark(x₁)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(z0)) → a__c(f(g(f(z0))))
a__f(z0) → f(z0)
a__c(z0) → d(z0)
a__c(z0) → c(z0)
a__h(z0) → a__c(d(z0))
a__h(z0) → h(z0)
mark(f(z0)) → a__f(mark(z0))
mark(c(z0)) → a__c(z0)
mark(h(z0)) → a__h(mark(z0))
mark(g(z0)) → g(z0)
mark(d(z0)) → d(z0)

Tuples:

MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

S tuples:none
K tuples:

MARK(f(z0)) → c7(A__F(mark(z0)), MARK(z0))
MARK(h(z0)) → c9(A__H(mark(z0)), MARK(z0))

Defined Rule Symbols:

a__f, a__c, a__h, mark

Defined Pair Symbols:

MARK

Compound Symbols:

c7, c9

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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