-- Mathlib/Analysis/Normed/Module/FiniteDimension.lean
theorem LipschitzOnWith.extend_finite_dimension {α : Type*} [PseudoMetricSpace α] {E' : Type*}
    [NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] {s : Set α} {f : α → E'}
    {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
    ∃ g : α → E', LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Proof state:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E'

Result:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Proof state:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
⊢ LipschitzWith ‖↑A‖₊ ⇑A

Tactic:
apply A.lipschitz

Result:
no goals

Proof state:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
have LA : LipschitzWith ‖A.toContinuousLinearMap‖₊ A := by apply A.lipschitz

Result:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Proof state:
case intro.intro
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
refine ⟨A.symm ∘ g, ?_, ?_⟩

Result:
case intro.intro.refine_1
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (⇑A.symm ∘ g)

case intro.intro.refine_2
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ EqOn f (⇑A.symm ∘ g) s

Proof state:
case intro.intro.refine_1
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
LAsymm : LipschitzWith ‖↑A.symm‖₊ ⇑A.symm
⊢ ‖↑A.symm‖₊ * ‖↑A‖₊ * K ≤
    (let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv;
      ‖↑A.symm‖₊ * ‖↑A‖₊ ⊔ 1) *
      K

Tactic:
exact mul_le_mul' (le_max_left _ _) le_rfl

Result:
no goals

Proof state:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ LipschitzWith ‖↑A.symm‖₊ ⇑A.symm

Tactic:
apply A.symm.lipschitz

Result:
no goals

Proof state:
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
obtain ⟨g, hg, gs⟩ :
  ∃ g : α → ι → ℝ, LipschitzWith (‖A.toContinuousLinearMap‖₊ * K) g ∧ EqOn (A ∘ f) g s :=
  L.extend_pi

Result:
case intro.intro
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s