-- Mathlib/Topology/TietzeExtension.lean
theorem exists_extension_norm_eq_of_isClosedEmbedding' (f : X →ᵇ ℝ) (e : C(X, Y))
    (he : IsClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f

Proof state:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
⊢ ∃ g, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f

Tactic:
set g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0

Result:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
⊢ ∃ g, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f

Proof state:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n => (g n).compContinuous e) atTop (𝓝 ((limUnder atTop g).compContinuous e))
hge : (limUnder atTop g).compContinuous e = f
⊢ ∃ g, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f

Tactic:
refine ⟨limUnder atTop g, le_antisymm ?_ ?_, hge⟩

Result:
case refine_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n => (g n).compContinuous e) atTop (𝓝 ((limUnder atTop g).compContinuous e))
hge : (limUnder atTop g).compContinuous e = f
⊢ ‖limUnder atTop g‖ ≤ ‖f‖

case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n => (g n).compContinuous e) atTop (𝓝 ((limUnder atTop g).compContinuous e))
hge : (limUnder atTop g).compContinuous e = f
⊢ ‖f‖ ≤ ‖limUnder atTop g‖

Proof state:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist ((F f).compContinuous e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - (g n).compContinuous e)
hgf : ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖
n : ℕ
⊢ dist (g n) (g (n + 1)) = ‖F (f - (g n).compContinuous e)‖

Tactic:
rw [g_succ, dist_eq_norm', add_sub_cancel_left]

Result:
no goals