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:

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Types:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false

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

Heuristically decided to analyse the following defined symbols:
f, activate

They will be analysed ascendingly in the following order:
f = activate

(8) Obligation:

TRS:
Rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Types:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false

Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))

The following defined symbols remain to be analysed:
activate, f

They will be analysed ascendingly in the following order:
f = activate

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_c:n__true:n__f:false2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
activate(gen_c:n__true:n__f:false2_0(+(1, 0)))

Induction Step:
activate(gen_c:n__true:n__f:false2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
f(activate(gen_c:n__true:n__f:false2_0(+(1, n4_0)))) →_IH
f(*3_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Types:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false

Lemmas:
activate(gen_c:n__true:n__f:false2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))

The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
f = activate

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Types:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false

Lemmas:
activate(gen_c:n__true:n__f:false2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_c:n__true:n__f:false2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Types:
f :: c:n__true:n__f:false → c:n__true:n__f:false
if :: c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false → c:n__true:n__f:false
c :: c:n__true:n__f:false
n__f :: c:n__true:n__f:false → c:n__true:n__f:false
n__true :: c:n__true:n__f:false
true :: c:n__true:n__f:false
false :: c:n__true:n__f:false
activate :: c:n__true:n__f:false → c:n__true:n__f:false
hole_c:n__true:n__f:false1_0 :: c:n__true:n__f:false
gen_c:n__true:n__f:false2_0 :: Nat → c:n__true:n__f:false

Lemmas:
activate(gen_c:n__true:n__f:false2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_c:n__true:n__f:false2_0(0) ⇔ n__true
gen_c:n__true:n__f:false2_0(+(x, 1)) ⇔ n__f(gen_c:n__true:n__f:false2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_c:n__true:n__f:false2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(18) BOUNDS(n^1, INF)