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 SlicingProof (LOWER BOUND(ID), 0 ms)
    ↳6 CpxRelTRS
    ↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳10 typed CpxTrs
    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 594 ms)
    ↳12 BEST
        ↳13 typed CpxTrs
            ↳14 LowerBoundsProof (⇔, 0 ms)
            ↳15 BOUNDS(n^1, INF)
        ↳16 typed CpxTrs
            ↳17 LowerBoundsProof (⇔, 0 ms)
            ↳18 BOUNDS(n^1, INF)

(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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
a__c/0
g/0
d/0
c/0

(6) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

Types:
a__f :: f:d:c:h:g → f:d:c:h:g
f :: f:d:c:h:g → f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g → f:d:c:h:g
mark :: f:d:c:h:g → f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g → f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat → f:d:c:h:g

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mark

(10) Obligation:

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

Types:
a__f :: f:d:c:h:g → f:d:c:h:g
f :: f:d:c:h:g → f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g → f:d:c:h:g
mark :: f:d:c:h:g → f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g → f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat → f:d:c:h:g

Generator Equations:
gen_f:d:c:h:g2_0(0) ⇔ d
gen_f:d:c:h:g2_0(+(x, 1)) ⇔ f(gen_f:d:c:h:g2_0(x))

The following defined symbols remain to be analysed:
mark

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_f:d:c:h:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Induction Base:
mark(gen_f:d:c:h:g2_0(+(1, 0)))

Induction Step:
mark(gen_f:d:c:h:g2_0(+(1, +(n4_0, 1)))) →RΩ(1)
a__f(mark(gen_f:d:c:h:g2_0(+(1, n4_0)))) →IH
a__f(*3_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
a__f :: f:d:c:h:g → f:d:c:h:g
f :: f:d:c:h:g → f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g → f:d:c:h:g
mark :: f:d:c:h:g → f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g → f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat → f:d:c:h:g

Lemmas:
mark(gen_f:d:c:h:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_f:d:c:h:g2_0(0) ⇔ d
gen_f:d:c:h:g2_0(+(x, 1)) ⇔ f(gen_f:d:c:h:g2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_f:d:c:h:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
a__f :: f:d:c:h:g → f:d:c:h:g
f :: f:d:c:h:g → f:d:c:h:g
a__c :: f:d:c:h:g
d :: f:d:c:h:g
a__h :: f:d:c:h:g → f:d:c:h:g
mark :: f:d:c:h:g → f:d:c:h:g
c :: f:d:c:h:g
h :: f:d:c:h:g → f:d:c:h:g
g :: f:d:c:h:g
hole_f:d:c:h:g1_0 :: f:d:c:h:g
gen_f:d:c:h:g2_0 :: Nat → f:d:c:h:g

Lemmas:
mark(gen_f:d:c:h:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_f:d:c:h:g2_0(0) ⇔ d
gen_f:d:c:h:g2_0(+(x, 1)) ⇔ f(gen_f:d:c:h:g2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_f:d:c:h:g2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(18) BOUNDS(n^1, INF)