The Runtime Complexity (innermost) 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(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

f(0') → true
f(1') → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(0') → true
f(1') → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c → true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c → 0':1':s:c
if :: true:false → 0':1':s:c → 0':1':s:c → 0':1':s:c
g :: 0':1':s:c → 0':1':s:c → 0':1':s:c
c :: 0':1':s:c → 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat → 0':1':s:c

(5) OrderProof (LOWER BOUND(ID) transformation)

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

They will be analysed ascendingly in the following order:
f < g

(6) Obligation:

Innermost TRS:
Rules:
f(0') → true
f(1') → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c → true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c → 0':1':s:c
if :: true:false → 0':1':s:c → 0':1':s:c → 0':1':s:c
g :: 0':1':s:c → 0':1':s:c → 0':1':s:c
c :: 0':1':s:c → 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat → 0':1':s:c

Generator Equations:
gen_0':1':s:c3_0(0) ⇔ 0'
gen_0':1':s:c3_0(+(x, 1)) ⇔ s(gen_0':1':s:c3_0(x))

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

They will be analysed ascendingly in the following order:
f < g

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_0':1':s:c3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
f(gen_0':1':s:c3_0(0)) →_R^Ω(1)
true

Induction Step:
f(gen_0':1':s:c3_0(+(n5_0, 1))) →_R^Ω(1)
f(gen_0':1':s:c3_0(n5_0)) →_IH
true

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
f(0') → true
f(1') → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c → true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c → 0':1':s:c
if :: true:false → 0':1':s:c → 0':1':s:c → 0':1':s:c
g :: 0':1':s:c → 0':1':s:c → 0':1':s:c
c :: 0':1':s:c → 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat → 0':1':s:c

Lemmas:
f(gen_0':1':s:c3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':1':s:c3_0(0) ⇔ 0'
gen_0':1':s:c3_0(+(x, 1)) ⇔ s(gen_0':1':s:c3_0(x))

The following defined symbols remain to be analysed:
g

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
f(0') → true
f(1') → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c → true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c → 0':1':s:c
if :: true:false → 0':1':s:c → 0':1':s:c → 0':1':s:c
g :: 0':1':s:c → 0':1':s:c → 0':1':s:c
c :: 0':1':s:c → 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat → 0':1':s:c

Lemmas:
f(gen_0':1':s:c3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':1':s:c3_0(0) ⇔ 0'
gen_0':1':s:c3_0(+(x, 1)) ⇔ s(gen_0':1':s:c3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_0':1':s:c3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
f(0') → true
f(1') → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c → true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c → 0':1':s:c
if :: true:false → 0':1':s:c → 0':1':s:c → 0':1':s:c
g :: 0':1':s:c → 0':1':s:c → 0':1':s:c
c :: 0':1':s:c → 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat → 0':1':s:c

Lemmas:
f(gen_0':1':s:c3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':1':s:c3_0(0) ⇔ 0'
gen_0':1':s:c3_0(+(x, 1)) ⇔ s(gen_0':1':s:c3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_0':1':s:c3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)