mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«2».«0»
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ {a, b, c}
h_not_col' : ¬Collinear ℝ {a', b', c'}
h : ∠ a b c = ∠ a' b' c'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist c a = k * dist c' a'

Tactic:
grind [dist_comm]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hp₄ : p₄ ∈ S
s_isom : ↥S →ᵃⁱ[ℝ] P := S.subtypeₐᵢ
p₁' : ↥S := ⟨p₁, ⋯⟩
p' : ↥S := ⟨p, ⋯⟩
p₃' : ↥S := ⟨p₃, ⋯⟩
p₂' : ↥S := ⟨p₂, hp₂⟩
p₄' : ↥S := ⟨p₄, hp₄⟩
h_dist' : dist p₁' p' * dist p₂' p' = dist p₃' p' * dist p₄' p'
hp₁'p₂' : ∠ p₁' p' p₂' = π
hp₃'p₄' : ∠ p₃' p' p₄' = π
hf2 : Fact (finrank ℝ ↥S.direction = 2)
this : Oriented ℝ (↥S.direction) (Fin 2) :=
  { positiveOrientation := (finBasisOfFinrankEq ℝ ↥S.direction ⋯).orientation }
⊢ AffineIndependent ℝ ![p₁', p', p₃']

Tactic:
← s_isom.toAffineMap.affineIndependent_iff s_isom.injective

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hp₄ : p₄ ∈ S
s_isom : ↥S →ᵃⁱ[ℝ] P := S.subtypeₐᵢ
p₁' : ↥S := ⟨p₁, ⋯⟩
p' : ↥S := ⟨p, ⋯⟩
p₃' : ↥S := ⟨p₃, ⋯⟩
p₂' : ↥S := ⟨p₂, hp₂⟩
p₄' : ↥S := ⟨p₄, hp₄⟩
h_dist' : dist p₁' p' * dist p₂' p' = dist p₃' p' * dist p₄' p'
hp₁'p₂' : ∠ p₁' p' p₂' = π
hp₃'p₄' : ∠ p₃' p' p₄' = π
hf2 : Fact (finrank ℝ ↥S.direction = 2)
this : Oriented ℝ (↥S.direction) (Fin 2) :=
  { positiveOrientation := (finBasisOfFinrankEq ℝ ↥S.direction ⋯).orientation }
⊢ AffineIndependent ℝ (⇑s_isom.toAffineMap ∘ ![p₁', p', p₃'])
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«2».«1»
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ {a, b, c}
h_not_col' : ¬Collinear ℝ {a', b', c'}
h : ∠ a b c = ∠ a' b' c'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist c b = k * dist c' b'

Tactic:
grind [dist_comm]

Goal state after:
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
h' :
  ∀ (i₂ : Fin (n + 3)),
    i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈ (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)
hi : p -ᵥ s.mongePoint ∈ (vectorSpan ℝ (Set.range s.points))ᗮ
hu : ⋃ i, {s.points i₁ -ᵥ s.points ↑i} = (fun x ↦ s.points i₁ -ᵥ x) '' s.points '' (Set.univ \ {i₁})
⊢ p -ᵥ s.mongePoint ∈ vectorSpan ℝ (Set.range s.points)

Tactic:
let s₁ : Finset (Fin (n + 3)) := univ.erase i₁

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
h' :
  ∀ (i₂ : Fin (n + 3)),
    i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈ (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)
hi : p -ᵥ s.mongePoint ∈ (vectorSpan ℝ (Set.range s.points))ᗮ
hu : ⋃ i, {s.points i₁ -ᵥ s.points ↑i} = (fun x ↦ s.points i₁ -ᵥ x) '' s.points '' (Set.univ \ {i₁})
s₁ : Finset (Fin (n + 3)) := univ.erase i₁
⊢ p -ᵥ s.mongePoint ∈ vectorSpan ℝ (Set.range s.points)
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

Goal state before:
pp: case refine_1
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p' : P
hp' : dist p' s.center < s.radius
hp : p' ∈ s
⊢ False

Tactic:
rw [mem_sphere] at hp

Goal state after:
case refine_1
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p' : P
hp' : dist p' s.center < s.radius
hp : dist p' s.center = s.radius
⊢ False
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ : Fin 3
⊢ t.orthocenter ∈ altitude t i₁

Tactic:
obtain ⟨i₂, i₃, h₁₂, h₂₃, h₁₃⟩ : ∃ i₂ i₃, i₁ ≠ i₂ ∧ i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
    decide +revert
  rw [orthocenter_eq_mongePoint, t.altitude_eq_mongePlane h₁₂ h₁₃ h₂₃]
  exact t.mongePoint_mem_mongePlane

Goal state after:
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case pos.inr
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h2 : c' = b'
h_dist_cb : c = b
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
simp_rw [h_dist_cb, h2, angle_self_right]

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.SmoothFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
H : Type u_4
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
N : Type u_6
inst✝⁸ : TopologicalSpace N
inst✝⁷ : ChartedSpace H N
E'' : Type u_7
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_8
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
N' : Type u_9
inst✝³ : TopologicalSpace N'
inst✝² : ChartedSpace H'' N'
n : ℕ∞ω
V : Type u_10
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace 𝕜 V
f : C^n⟮I, N; 𝓘(𝕜, V), V⟯
x : N
⊢ (0 • f) x = 0 x

Tactic:
exact zero_smul _ _

Goal state after:
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case right.inr.inl
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
ho : t.orthocenter ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)
hpi : Function.Injective p
i₁ i₂ i₃ j₂ j₃ : Fin 3
left✝² : i₁ ≠ i₂
left✝¹ : i₁ ≠ i₃
left✝ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : p i₁ = t.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t.points j₂ = p i₂
h₃ : t.points j₃ = p i₃
⊢ dist (p i₂) ((reflection (affineSpan ℝ (t.points '' {j₂, j₃}))) (Simplex.circumcenter t)) = Simplex.circumradius t

pp: case right.inr.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
ho : t.orthocenter ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)
hpi : Function.Injective p
i₁ i₂ i₃ j₂ j₃ : Fin 3
left✝² : i₁ ≠ i₂
left✝¹ : i₁ ≠ i₃
left✝ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : p i₁ = t.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t.points j₂ = p i₂
h₃ : t.points j₃ = p i₃
⊢ dist (p i₃) ((reflection (affineSpan ℝ (t.points '' {j₂, j₃}))) (Simplex.circumcenter t)) = Simplex.circumradius t

Tactic:
· rw [← h₂, dist_reflection_eq_of_mem _
       (mem_affineSpan ℝ (Set.mem_image_of_mem _ (Set.mem_insert _ _)))]
      exact t.dist_circumcenter_eq_circumradius _

Goal state after:
case right.inr.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
ho : t.orthocenter ∉ Set.range t.points
p : Fin 3 → P
hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)
hpi : Function.Injective p
i₁ i₂ i₃ j₂ j₃ : Fin 3
left✝² : i₁ ≠ i₂
left✝¹ : i₁ ≠ i₃
left✝ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : p i₁ = t.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t.points j₂ = p i₂
h₃ : t.points j₃ = p i₃
⊢ dist (p i₃) ((reflection (affineSpan ℝ (t.points '' {j₂, j₃}))) (Simplex.circumcenter t)) = Simplex.circumradius t
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
ho : OrthocentricSystem s
t : Triangle ℝ P
ht : Set.range t.points ⊆ s
⊢ s = insert t.orthocenter (Set.range t.points)

Tactic:
rcases ho with ⟨t₀, ht₀o, ht₀s⟩

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
t : Triangle ℝ P
ht : Set.range t.points ⊆ s
t₀ : Triangle ℝ P
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.range t₀.points)
⊢ s = insert t.orthocenter (Set.range t.points)
Module: Mathlib.Geometry.Manifold.LocalDiffeomorph

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
F : Type u_3
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
F' : Type u_4
inst✝¹⁰ : NormedAddCommGroup F'
inst✝⁹ : NormedSpace 𝕜 F'
H₁ : Type u_5
inst✝⁸ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁷ : TopologicalSpace H₂
H₃ : Type u_7
inst✝⁶ : TopologicalSpace H₃
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
K : ModelWithCorners 𝕜 F' H₃
M : Type u_8
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H₁ M
N : Type u_9
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace H₂ N
P : Type u_10
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace H₃ P
n : WithTop ℕ∞
f : M → N
s : Set M
x : M
I' : ?m.25
hf : IsLocalDiffeomorphAt I J n f x
hn : n ≠ 0
⊢ Function.LeftInverse ⇑(mfderiv% ↑hf.localInverse.toPartialEquiv (f x)) ⇑(mfderiv% f x)

Tactic:
by
    apply ContinuousLinearMap.leftInverse_of_comp
    rw [← mfderiv_id, ← hf.localInverse_eventuallyEq_left.mfderiv_eq]
    exact (mfderiv_comp _ (hf.localInverse_mdifferentiableAt hn) (hf.mdifferentiableAt hn)).symm

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
N : Type u_7
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
n : WithTop ℕ∞
hn : n ≠ 0
f : M → N
x : M
hf : IsLocalDiffeomorphAt I I' n f x
h : I'.IsInteriorPoint (f x)
⊢ I.IsInteriorPoint x

Tactic:
← hf.localInverse_left_inv hf.localInverse_mem_target

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
N : Type u_7
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
n : WithTop ℕ∞
hn : n ≠ 0
f : M → N
x : M
hf : IsLocalDiffeomorphAt I I' n f x
h : I'.IsInteriorPoint (f x)
⊢ I.IsInteriorPoint (↑hf.localInverse.toPartialEquiv (f x))
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
N : Type u_7
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
h : ModelWithCorners.boundary (M × N) =
  (ModelWithCorners.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (ModelWithCorners.interior N)ᶜ :=
  Trans.trans
    (Trans.trans (Trans.trans rfl (Eq.symm compl_interior))
      (Eq.mpr
        (id (congrArg (fun _a ↦ _aᶜ = (ModelWithCorners.interior M ×ˢ ModelWithCorners.interior N)ᶜ) interior_prod))
        (Eq.refl (ModelWithCorners.interior M ×ˢ ModelWithCorners.interior N)ᶜ)))
    (Eq.mpr
      (id
        (congrArg (fun _a ↦ _a = (ModelWithCorners.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (ModelWithCorners.interior N)ᶜ)
          (compl_prod_eq_union (ModelWithCorners.interior M) (ModelWithCorners.interior N))))
      (Eq.refl ((ModelWithCorners.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (ModelWithCorners.interior N)ᶜ)))
⊢ ModelWithCorners.boundary M ×ˢ univ ∪ univ ×ˢ ModelWithCorners.boundary N =
    univ.prod (ModelWithCorners.boundary N) ∪ (ModelWithCorners.boundary M).prod univ

Tactic:
union_comm

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
N : Type u_7
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
h : ModelWithCorners.boundary (M × N) =
  (ModelWithCorners.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (ModelWithCorners.interior N)ᶜ :=
  Trans.trans
    (Trans.trans (Trans.trans rfl (Eq.symm compl_interior))
      (Eq.mpr
        (id (congrArg (fun _a ↦ _aᶜ = (ModelWithCorners.interior M ×ˢ ModelWithCorners.interior N)ᶜ) interior_prod))
        (Eq.refl (ModelWithCorners.interior M ×ˢ ModelWithCorners.interior N)ᶜ)))
    (Eq.mpr
      (id
        (congrArg (fun _a ↦ _a = (ModelWithCorners.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (ModelWithCorners.interior N)ᶜ)
          (compl_prod_eq_union (ModelWithCorners.interior M) (ModelWithCorners.interior N))))
      (Eq.refl ((ModelWithCorners.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (ModelWithCorners.interior N)ᶜ)))
⊢ univ ×ˢ ModelWithCorners.boundary N ∪ ModelWithCorners.boundary M ×ˢ univ =
    univ.prod (ModelWithCorners.boundary N) ∪ (ModelWithCorners.boundary M).prod univ
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
x₀ : B
ι : Type u_9
s : Finset ι
t : ι → (x : B) → E x
hs : ∀ i ∈ s, MDiffAt[u] (T% t) x₀
⊢ (MDiffAt[u] fun x ↦ ⟨x, ∑ i ∈ s, t i x⟩) x₀

Tactic:
classical
  induction s using Finset.induction_on with
  | empty => simpa using! (contMDiffWithinAt_zeroSection 𝕜 E).mdifferentiableWithinAt one_ne_zero
  | insert i s hi h =>
    simp only [Finset.mem_insert, forall_eq_or_imp] at hs
    simpa [Finset.sum_insert hi] using mdifferentiableWithinAt_add_section (hs.1) (h hs.2)

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
s : (x : B) → E x
ψ : B → 𝕜
hψ : MDiff[u] ψ
ht : IsOpen[inst✝¹³] u
ht' : tsupport ψ ⊆ u
hs : MDiff[u] (T% s)
⊢ ∀ y ∈ (tsupport ψ)ᶜ, ⟨y, (ψ • s) y⟩ = zeroSection F E y

Tactic:
intro y hy

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
s : (x : B) → E x
ψ : B → 𝕜
hψ : MDiff[u] ψ
ht : IsOpen[inst✝¹³] u
ht' : tsupport ψ ⊆ u
hs : MDiff[u] (T% s)
y : B
hy : y ∈ (tsupport ψ)ᶜ
⊢ ⟨y, (ψ • s) y⟩ = zeroSection F E y
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
u : Set B
s : (x : B) → E x
ψ : B → 𝕜
hψ : MDiff[u] ψ
ht : IsOpen[inst✝¹³] u
ht' : tsupport ψ ⊆ u
hs : MDiff[u] (T% s)
⊢ MDiff[(tsupport ψ)ᶜ] fun x ↦ ⟨x, (ψ • s) x⟩

Tactic:
apply ((mdifferentiable_zeroSection _ _).mdifferentiableOn (s := (tsupport ψ)ᶜ)).congr
    intro y hy
    simp [image_eq_zero_of_notMem_tsupport hy, zeroSection]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.Hom

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
inst✝¹⁷ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
EB : Type u_4
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
E₁ : B → Type u_6
inst✝¹¹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁰ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝⁶ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁵ : ∀ (x : B), IsTopologicalAddGroup (E₁ x)
inst✝⁴ : ∀ (x : B), ContinuousSMul 𝕜 (E₁ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : ContMDiffVectorBundle n F₁ E₁ IB
e : Trivialization F₁ TotalSpace.proj
inst✝ : MemTrivializationAtlas e
x : B
hx : x ∈ e.baseSet
hx' : x ∈ (trivializationAt F₁ E₁ x).baseSet
⊢ ContMDiffAt IB 𝓘(𝕜, F₁ →L[𝕜] F₁) n
    (fun x_1 ↦ (↑(trivializationAt (F₁ →L[𝕜] F₁) (fun x ↦ F₁ →L[𝕜] E₁ x) ⟨x, symmL 𝕜 e x⟩.proj) ⟨x_1, symmL 𝕜 e x_1⟩).2)
    x

Tactic:
apply (contMDiffAt_coordChangeL hx hx').congr_of_eventuallyEq

Goal state after:
𝕜 : Type u_1
B : Type u_2
F₁ : Type u_3
inst✝¹⁷ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
EB : Type u_4
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹³ : TopologicalSpace B
inst✝¹² : ChartedSpace HB B
E₁ : B → Type u_6
inst✝¹¹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁰ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝⁶ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁵ : ∀ (x : B), IsTopologicalAddGroup (E₁ x)
inst✝⁴ : ∀ (x : B), ContinuousSMul 𝕜 (E₁ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : ContMDiffVectorBundle n F₁ E₁ IB
e : Trivialization F₁ TotalSpace.proj
inst✝ : MemTrivializationAtlas e
x : B
hx : x ∈ e.baseSet
hx' : x ∈ (trivializationAt F₁ E₁ x).baseSet
⊢ (fun x_1 ↦
      (↑(trivializationAt (F₁ →L[𝕜] F₁) (fun x ↦ F₁ →L[𝕜] E₁ x) ⟨x, symmL 𝕜 e x⟩.proj) ⟨x_1, symmL 𝕜 e x_1⟩).2) =ᶠ[𝓝 x]
    fun b ↦ ↑(coordChangeL 𝕜 e (trivializationAt F₁ E₁ x) b)