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

(0) Obligation:

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

a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__if(false, X, if(false, X23713_0, X33714_0)) →⁺ a__if(false, mark(X23713_0), X33714_0)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X33714_0 / if(false, X23713_0, X33714_0)].
The result substitution is [X / mark(X23713_0)].

(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(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

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

Heuristically decided to analyse the following defined symbols:
a__f, a__if, mark

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__if = mark

(8) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

Generator Equations:
gen_c:true:f:false:if2_0(0) ⇔ c
gen_c:true:f:false:if2_0(+(x, 1)) ⇔ f(gen_c:true:f:false:if2_0(x))

The following defined symbols remain to be analysed:
a__if, a__f, mark

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__if = mark

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

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

(10) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

Generator Equations:
gen_c:true:f:false:if2_0(0) ⇔ c
gen_c:true:f:false:if2_0(+(x, 1)) ⇔ f(gen_c:true:f:false:if2_0(x))

The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__if = mark

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

Proved the following rewrite lemma:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

Induction Base:
mark(gen_c:true:f:false:if2_0(+(1, 0)))

Induction Step:
mark(gen_c:true:f:false:if2_0(+(1, +(n26_0, 1)))) →_R^Ω(1)
a__f(mark(gen_c:true:f:false:if2_0(+(1, n26_0)))) →_IH
a__f(*3_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

Lemmas:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

Generator Equations:
gen_c:true:f:false:if2_0(0) ⇔ c
gen_c:true:f:false:if2_0(+(x, 1)) ⇔ f(gen_c:true:f:false:if2_0(x))

The following defined symbols remain to be analysed:
a__f, a__if

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__if = mark

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

Lemmas:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

Generator Equations:
gen_c:true:f:false:if2_0(0) ⇔ c
gen_c:true:f:false:if2_0(+(x, 1)) ⇔ f(gen_c:true:f:false:if2_0(x))

The following defined symbols remain to be analysed:
a__if

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__if = mark

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

Lemmas:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

Generator Equations:
gen_c:true:f:false:if2_0(0) ⇔ c
gen_c:true:f:false:if2_0(+(x, 1)) ⇔ f(gen_c:true:f:false:if2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

TRS:
Rules:
a__f(X) → a__if(mark(X), c, f(true))
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(f(X)) → a__f(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), mark(X2), X3)
mark(c) → c
mark(true) → true
mark(false) → false
a__f(X) → f(X)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if → c:true:f:false:if
a__if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
mark :: c:true:f:false:if → c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if → c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if → c:true:f:false:if → c:true:f:false:if → c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat → c:true:f:false:if

Lemmas:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

Generator Equations:
gen_c:true:f:false:if2_0(0) ⇔ c
gen_c:true:f:false:if2_0(+(x, 1)) ⇔ f(gen_c:true:f:false:if2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) → *3_0, rt ∈ Ω(n26₀)

(22) BOUNDS(n^1, INF)