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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 92 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 123 ms)
    ↳10 typed CpxTrs

(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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
mark(f(X)) →+ a__f(mark(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / f(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative 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)

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
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)

Types:
a__f :: f:g:d:c:h → f:g:d:c:h
f :: f:g:d:c:h → f:g:d:c:h
a__c :: f:g:d:c:h → f:g:d:c:h
g :: f:g:d:c:h → f:g:d:c:h
d :: f:g:d:c:h → f:g:d:c:h
a__h :: f:g:d:c:h → f:g:d:c:h
mark :: f:g:d:c:h → f:g:d:c:h
c :: f:g:d:c:h → f:g:d:c:h
h :: f:g:d:c:h → f:g:d:c:h
hole_f:g:d:c:h1_0 :: f:g:d:c:h
gen_f:g:d:c:h2_0 :: Nat → f:g:d:c:h

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mark

(8) Obligation:

TRS:
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)

Types:
a__f :: f:g:d:c:h → f:g:d:c:h
f :: f:g:d:c:h → f:g:d:c:h
a__c :: f:g:d:c:h → f:g:d:c:h
g :: f:g:d:c:h → f:g:d:c:h
d :: f:g:d:c:h → f:g:d:c:h
a__h :: f:g:d:c:h → f:g:d:c:h
mark :: f:g:d:c:h → f:g:d:c:h
c :: f:g:d:c:h → f:g:d:c:h
h :: f:g:d:c:h → f:g:d:c:h
hole_f:g:d:c:h1_0 :: f:g:d:c:h
gen_f:g:d:c:h2_0 :: Nat → f:g:d:c:h

Generator Equations:
gen_f:g:d:c:h2_0(0) ⇔ hole_f:g:d:c:h1_0
gen_f:g:d:c:h2_0(+(x, 1)) ⇔ f(gen_f:g:d:c:h2_0(x))

The following defined symbols remain to be analysed:
mark

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol mark.

(10) Obligation:

TRS:
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)

Types:
a__f :: f:g:d:c:h → f:g:d:c:h
f :: f:g:d:c:h → f:g:d:c:h
a__c :: f:g:d:c:h → f:g:d:c:h
g :: f:g:d:c:h → f:g:d:c:h
d :: f:g:d:c:h → f:g:d:c:h
a__h :: f:g:d:c:h → f:g:d:c:h
mark :: f:g:d:c:h → f:g:d:c:h
c :: f:g:d:c:h → f:g:d:c:h
h :: f:g:d:c:h → f:g:d:c:h
hole_f:g:d:c:h1_0 :: f:g:d:c:h
gen_f:g:d:c:h2_0 :: Nat → f:g:d:c:h

Generator Equations:
gen_f:g:d:c:h2_0(0) ⇔ hole_f:g:d:c:h1_0
gen_f:g:d:c:h2_0(+(x, 1)) ⇔ f(gen_f:g:d:c:h2_0(x))

No more defined symbols left to analyse.