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

Goal state before:
pp: case neg
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
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
this :
  (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
⊢ ⅟2 •
      ((↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) +
        ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)

Tactic:
rw [this, ← two_smul ℝ, smul_smul]

Goal state after:
case neg
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
hn : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hn1 : ¬n = 1
hltn : 1 < n
hnsub1 : ↑(n - 1) = ↑n - 1
this :
  (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -ᵥ s.circumcenter) =
    ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
⊢ (⅟2 * 2) • ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter) = ((↑n + 1) / ↑n) • (s.centroid -ᵥ s.circumcenter)
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case ha
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 := ⋯
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
⊢ 0 < dist a b

pp: case hb
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 := ⋯
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
⊢ 0 < dist a' b'

Tactic:
· simp [dist_pos, ne₁₂_of_not_collinear h_not_col]

Goal state after:
case hb
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 := ⋯
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
⊢ 0 < dist a' b'
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 i₂ : Fin (n + 3)
⊢ s.mongePoint ∈ mk' (Finset.centroid ℝ {i₁, i₂}ᶜ s.points) (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ ∧
    s.mongePoint ∈ affineSpan ℝ (Set.range s.points)

Tactic:
← vsub_right_mem_direction_iff_mem (self_mem_mk' _ _),

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 i₂ : Fin (n + 3)
⊢ s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points ∈
      (mk' (Finset.centroid ℝ {i₁, i₂}ᶜ s.points) (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ).direction ∧
    s.mongePoint ∈ affineSpan ℝ (Set.range s.points)
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
⊢ ↑(extChartAt I x) x ∉ interior (range ↑I) →
    ↑(extChartAt I x) x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range ↑I) \ interior (range ↑I)

Tactic:
I.isClosed_range.closure_eq,

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
⊢ ↑(extChartAt I x) x ∉ interior (range ↑I) → ↑(extChartAt I x) x ∈ range ↑I \ interior (range ↑I)
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'
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
⊢ I.IsInteriorPoint x ↔ I'.IsInteriorPoint (f x)

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩

Goal state after:
case refine_1
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 x
⊢ I'.IsInteriorPoint (f x)

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
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'
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
s : Set M
hf : IsLocalDiffeomorphOn I I' n f s
⊢ f ⁻¹' ModelWithCorners.interior N ∩ s = ModelWithCorners.interior M ∩ s

Tactic:
by
  ext x
  simpa using! fun hx ↦ ((hf ⟨x, hx⟩).isInteriorPoint_iff hn).symm

Goal state after:
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'
p : M × N
⊢ interior (range ↑I ×ˢ range ↑J) = interior (range ↑(I.prod J))

Tactic:
← range_prodMap,

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'
p : M × N
⊢ interior (range (Prod.map ↑I ↑J)) = interior (range ↑(I.prod J))
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 × N) = univ.prod (ModelWithCorners.boundary N) ∪ (ModelWithCorners.boundary M).prod univ

Tactic:
rw [h, I.compl_interior, J.compl_interior, 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.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
inst✝ : BoundarylessManifold J N
⊢ univ.prod (ModelWithCorners.boundary N) ∪ (ModelWithCorners.boundary M).prod univ =
    (ModelWithCorners.boundary M).prod univ

Tactic:
Boundaryless.boundary_eq_empty (I := J)

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
inst✝ : BoundarylessManifold J N
⊢ univ.prod ∅ ∪ (ModelWithCorners.boundary M).prod univ = (ModelWithCorners.boundary M).prod univ
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
inst✝ : BoundarylessManifold J N
this : univ.prod ∅ = ∅
⊢ univ.prod ∅ ∪ (ModelWithCorners.boundary M).prod univ = (ModelWithCorners.boundary M).prod univ

Tactic:
rw [this, empty_union]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
X : LeftInvariantDerivation I G
f : C^∞⟮I, G; 𝕜⟯⟨g * h⟩
⊢ ((evalAt (g * h)) X) f = ((𝒅ₕ ⋯) ((evalAt h) X)) f

Tactic:
← left_invariant,

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g h : G
X : LeftInvariantDerivation I G
f : C^∞⟮I, G; 𝕜⟯⟨g * h⟩
⊢ ((𝒅ₕ ⋯) ((evalAt 1) X)) f = ((𝒅ₕ ⋯) ((evalAt h) X)) f
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup F
inst✝¹⁹ : NormedSpace 𝕜 F
inst✝¹⁸ : TopologicalSpace (TotalSpace F E)
inst✝¹⁷ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝¹³ : NormedAddCommGroup EM
inst✝¹² : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹¹ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace HM M
inst✝⁸ : TopologicalSpace B
inst✝⁷ : ChartedSpace HB B
inst✝⁶ : FiberBundle F E
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module 𝕜 (E x)
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle 1 F E IB
e : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
e' : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e'
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : MDiffAt[s] (TotalSpace.proj ∘ f) x₀
he'f : (MDiffAt[s] fun x ↦ (↑e (f x)).2) x₀
he : (f x₀).proj ∈ e.baseSet
he' : (f x₀).proj ∈ e'.baseSet
⊢ (MDiffAt[s] fun x ↦ (↑e' (f x)).2) x₀

Tactic:
refine (hf.coordChange he'f he he').congr_of_eventuallyEq ?_ (by simp [he])

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup F
inst✝¹⁹ : NormedSpace 𝕜 F
inst✝¹⁸ : TopologicalSpace (TotalSpace F E)
inst✝¹⁷ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁶ : NormedAddCommGroup EB
inst✝¹⁵ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝¹³ : NormedAddCommGroup EM
inst✝¹² : NormedSpace 𝕜 EM
HM : Type u_11
inst✝¹¹ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace HM M
inst✝⁸ : TopologicalSpace B
inst✝⁷ : ChartedSpace HB B
inst✝⁶ : FiberBundle F E
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module 𝕜 (E x)
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle 1 F E IB
e : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
e' : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e'
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : MDiffAt[s] (TotalSpace.proj ∘ f) x₀
he'f : (MDiffAt[s] fun x ↦ (↑e (f x)).2) x₀
he : (f x₀).proj ∈ e.baseSet
he' : (f x₀).proj ∈ e'.baseSet
⊢ (fun x ↦ (↑e' (f x)).2) =ᶠ[𝓝[s] x₀] fun y ↦ e.coordChange e' ((TotalSpace.proj ∘ f) y) (↑e (f y)).2
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set M
x : M
a : M → 𝕜
ha : MDiffAt[s] a x
g : M → F
hg : MDiffAt[s] g x
hs : UniqueMDiffAt[s] x
⊢ d[s] (a • g) x = a x • d[s] g x + (d[s] a x).smulRight (g x)

Tactic:
by
  refine HasMFDerivWithinAt.mfderivWithin ⟨ha.1.smul hg.1, ?_⟩ hs
  convert! ha.hasMFDerivWithinAt.2.smul hg.hasMFDerivWithinAt.2
  simp
  rfl

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
E' : Type u_5
inst✝¹ : NormedAddCommGroup E'
inst✝ : NormedSpace 𝕜 E'
x : M
f : M → E'
h : MDiffAt f x
⊢ d% f x = fderivWithin 𝕜 (writtenInExtChartAt I 𝓘(𝕜, E') x f) (range ↑I) (↑(extChartAt I x) x)

Tactic:
by
  convert! h.mfderiv

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J 1 N
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiff[u]
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x ↦ f x (g x)) t x₀
h4f : (fun x ↦ f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝[t ×ˢ u] (x₀, g x₀), ContMDiffWithinAt (J.prod I) I' 1 (uncurry f) (t ×ˢ u) x'
h2f : ∀ᶠ (x₂ : N) in 𝓝[t] x₀, ContMDiffWithinAt I I' 1 (f x₂) u (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝[t] x₀
this :
  ContDiffWithinAt 𝕜 m
    (fun x ↦
      fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm)
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u)
        (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x))))
    (↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)
⊢ ContDiffWithinAt 𝕜 m
    (fun x ↦
      fderivWithin 𝕜
        (fun x_1 ↦ ↑(extChartAt I' (f x₀ (g x₀))) (f (↑(extChartAt J x₀).symm x) (↑(extChartAt I (g x₀)).symm x_1)))
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u)
        (↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x))))
    (↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)

Tactic:
exact this

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case neg
x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ↑(Icc x y)
φ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) :=
  ContinuousLinearEquiv.prodUnique ℝ (EuclideanSpace ℝ (Fin 1)) Unit
φ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := φ₀.trans (PiLp.equivOfUnique 2 ℝ fun x ↦ ℝ)
hz : ¬↑z < y
⊢ IsImmersionAtOfComplement Unit (𝓡∂ 1) 𝓘(ℝ, ℝ) n (fun z ↦ ↑z) z

Tactic:
apply IsImmersionAtOfComplement.mk_of_continuousAt (by fun_prop) φ
      (chartAt (EuclideanHalfSpace 1) z)
      (Homeomorph.pointReflection (y / 2)).toOpenPartialHomeomorph (mem_chart_source _ z)
      (by simp [Homeomorph.pointReflection]) (chart_mem_maximalAtlas _) ?_

Goal state after:
case neg
x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ↑(Icc x y)
φ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) := ⋯
φ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := ⋯
hz : ¬↑z < y
⊢ EqOn
    (↑((Homeomorph.pointReflection (y / 2)).toOpenPartialHomeomorph.extend 𝓘(ℝ, ℝ)) ∘
      (fun z ↦ ↑z) ∘ ↑((chartAt (EuclideanHalfSpace 1) z).extend (𝓡∂ 1)).symm)
    (⇑φ ∘ fun x ↦ (x, 0)) ((chartAt (EuclideanHalfSpace 1) z).extend (𝓡∂ 1)).target

x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ↑(Icc x y)
φ₀ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] EuclideanSpace ℝ (Fin 1) :=
  ContinuousLinearEquiv.prodUnique ℝ (EuclideanSpace ℝ (Fin 1)) Unit
φ : (EuclideanSpace ℝ (Fin 1) × Unit) ≃L[ℝ] ℝ := φ₀.trans (PiLp.equivOfUnique 2 ℝ fun x ↦ ℝ)
hz : ¬↑z < y
⊢ (Homeomorph.pointReflection (y / 2)).toOpenPartialHomeomorph ∈ maximalAtlas 𝓘(ℝ, ℝ) n ℝ
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ℝ
hz : z ∈ Icc x y
h'z : ↑(projIcc x y ⋯ z) < y
this✝ :
  ContDiff ℝ n fun w ↦
    have this := toLp 2 fun x_1 ↦ w - x;
    this
w : ℝ
hw : w ∈ Icc x y
i : Fin 1
this : max x (min y w) - x = w - x
⊢ (((Subtype.val ∘ ↑(IccLeftChart x y)) ∘ projIcc x y ⋯ ∘ ↑(chartAt ℝ z).symm) w).ofLp i =
    (have this := toLp 2 fun x_1 ↦ w - x;
        this).ofLp
      i

Tactic:
simpa [modelWithCornersEuclideanHalfSpace, IccLeftChart]

Goal state after: