-- Mathlib/Data/Real/Cardinality.lean
theorem increasing_cantorFunction (h1 : 0 < c) (h2 : c < 1 / 2) {n : ℕ} {f g : ℕ → Bool}
    (hn : ∀ k < n, f k = g k) (fn : f n = false) (gn : g n = true) :
    cantorFunction c f < cantorFunction c g

Proof state:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ cantorFunction c f_max < cantorFunction c g

Tactic:
refine lt_of_lt_of_le ?_ (cantorFunction_le (le_of_lt h1) h3 hg_min)

Result:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ cantorFunction c f_max < cantorFunction c g_min

Proof state:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ cantorFunction c f_max < cantorFunction c g_min

Tactic:
convert this

Result:
case h.e'_3
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ cantorFunction c f_max = c / (1 - c)

case h.e'_4
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ cantorFunction c g_min = 1

Proof state:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ 0 < 1 - c

Tactic:
rwa [sub_pos]

Result:
no goals

Proof state:
case h.e'_3
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ cantorFunction c f_max = c / (1 - c)

Tactic:
rw [cantorFunction_succ _ (le_of_lt h1) h3, div_eq_mul_inv, ←
  tsum_geometric_of_lt_one (le_of_lt h1) h3]

Result:
case h.e'_3
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ ((bif f_max 0 then 1 else 0) + c * cantorFunction c fun n => f_max (n + 1)) = c * ∑' (n : ℕ), c ^ n

Proof state:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
n : ℕ
f g : ℕ → Bool
hn : ∀ k < n, f k = g k
fn : f n = false
gn : g n = true
⊢ 1 / 2 < 1

Tactic:
norm_num

Result:
no goals

Proof state:
case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ ((bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <
    (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)

Tactic:
rw [hn 0 <| zero_lt_succ n]

Result:
case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ ((bif g 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <
    (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)

Proof state:
case h.e'_4
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ cantorFunction c g_min = 1

Tactic:
refine (tsum_eq_single 0 ?_).trans ?_

Result:
case h.e'_4.refine_1
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ ∀ (b' : ℕ), b' ≠ 0 → cantorFunctionAux c g_min b' = 0

case h.e'_4.refine_2
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
⊢ cantorFunctionAux c g_min 0 = 1

Proof state:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
n : ℕ
f g : ℕ → Bool
hn : ∀ k < n, f k = g k
fn : f n = false
gn : g n = true
⊢ cantorFunction c f < cantorFunction c g

Tactic:
have h3 : c < 1 := by
  apply h2.trans
  norm_num

Result:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
n : ℕ
f g : ℕ → Bool
hn : ∀ k < n, f k = g k
fn : f n = false
gn : g n = true
h3 : c < 1
⊢ cantorFunction c f < cantorFunction c g

Proof state:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
n : ℕ
hn : f n = true
⊢ f_max n = true

Tactic:
cases n

Result:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hn : f 0 = true
⊢ f_max 0 = true

case succ
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
n✝ : ℕ
hn : f (n✝ + 1) = true
⊢ f_max (n✝ + 1) = true

Proof state:
case succ.bc
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
n : ℕ
ih : ∀ {f g : ℕ → Bool}, (∀ k < n, f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g
f g : ℕ → Bool
hn : ∀ k < n + 1, f k = g k
fn : f (n + 1) = false
gn : g (n + 1) = true
⊢ (cantorFunction c fun n => f (n + 1)) < cantorFunction c fun n => g (n + 1)

Tactic:
exact ih (fun k hk => hn _ <| Nat.succ_lt_succ hk) fn gn

Result:
no goals