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

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
s : Sphere P
p : P
v : V
hp : 0 = (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v
hv : ¬v = 0
⊢ ⟪v, p -ᵥ s.center⟫ = 0

Tactic:
eq_comm,

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
s : Sphere P
p : P
v : V
hp : (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v = 0
hv : ¬v = 0
⊢ ⟪v, p -ᵥ s.center⟫ = 0
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.mongePlane i₁ i₂ = s.mongePlane i₂ i₁

Tactic:
simp_rw [mongePlane_def]
  congr 3
  · congr 1
    exact pair_comm _ _
  · ext
    simp_rw [Submodule.mem_span_singleton]
    constructor
    all_goals rintro ⟨r, rfl⟩; use -r; rw [neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev]

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

Goal state before:
pp: case refine_2
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 : p ∈ s
v : V
p' : P
h : p' ∈ s
r : ℝ
hr : r • v = p' -ᵥ p
⊢ p' = p ∨ p' = s.secondInter p v

Tactic:
eq_comm,

Goal state after:
case refine_2
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 : p ∈ s
v : V
p' : P
h : p' ∈ s
r : ℝ
hr : p' -ᵥ p = r • v
⊢ p' = p ∨ p' = s.secondInter p v
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)
⊢ ∀ u ∈ ℝ ∙ (s.points i₁ -ᵥ s.points i₂), ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, u⟫ = 0

Tactic:
intro v hv

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)
v : V
hv : v ∈ ℝ ∙ (s.points i₁ -ᵥ s.points i₂)
⊢ ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, v⟫ = 0
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p
he : p = s.secondInter p (p' -ᵥ p)
⊢ False

Tactic:
eq_comm,

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
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p
he : s.secondInter p (p' -ᵥ p) = p
⊢ False
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 }
hncol : ¬Collinear ℝ {p₁', p', p₃'}
⊢ Cospherical {p₁', p₂', p₃', p₄'}

Tactic:
exact cospherical_of_mul_dist_eq_mul_dist_of_angle_eq_pi_aux h_dist' hp₁'p₂' hp₃'p₄' hncol

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : I.Boundaryless
x : M
⊢ I.IsInteriorPoint x

Tactic:
let r := ((chartAt H x).isOpen_extend_target (I := I)).interior_eq
    have : extChartAt I x = (chartAt H x).extend I := rfl
    rw [← this] at r
    rw [isInteriorPoint_iff, r]
    exact PartialEquiv.map_source _ (mem_extChartAt_source _)

Goal state after:
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
hf : IsLocalDiffeomorph I J n f
hf' : Function.Bijective f
g : N → M
hgInverse : Function.LeftInverse g f ∧ Function.RightInverse g f
Φ : M → PartialDiffeomorph I J M N n
hyp : ∀ (x : M), x ∈ (Φ x).source ∧ EqOn f (↑(Φ x).toPartialEquiv) (Φ x).source
aux : ∀ (x : M), EqOn g (↑(Φ x).symm.toPartialEquiv) (Φ x).target
y : N
⊢ ContMDiffAt J I n (⇑{ toFun := f, invFun := g, left_inv := ⋯, right_inv := ⋯ }.symm) y

Tactic:
let x := g y

Goal state after:
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
hf : IsLocalDiffeomorph I J n f
hf' : Function.Bijective f
g : N → M
hgInverse : Function.LeftInverse g f ∧ Function.RightInverse g f
Φ : M → PartialDiffeomorph I J M N n
hyp : ∀ (x : M), x ∈ (Φ x).source ∧ EqOn f (↑(Φ x).toPartialEquiv) (Φ x).source
aux : ∀ (x : M), EqOn g (↑(Φ x).symm.toPartialEquiv) (Φ x).target
y : N
x : M := g y
⊢ ContMDiffAt J I n (⇑{ toFun := f, invFun := g, left_inv := ⋯, right_inv := ⋯ }.symm) y
Module: Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f✝ f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y : LeftInvariantDerivation I G
g : G
f : C^∞⟮I, G; 𝕜⟯⟨g⟩
hX : ((𝒅ₕ ⋯) ((Derivation.evalAt 1) ↑X)) (Y f) = ((Derivation.evalAt g) ↑X) (Y f)
hY : ((𝒅ₕ ⋯) ((Derivation.evalAt 1) ↑Y)) (X f) = ((Derivation.evalAt g) ↑Y) (X f)
⊢ ((𝒅ₕ ⋯) ((Derivation.evalAt 1) ⁅↑X, ↑Y⁆)) f = ((Derivation.evalAt g) ⁅↑X, ↑Y⁆) f

Tactic:
rw [hfdifferential_apply, fdifferential_apply, Derivation.evalAt_apply] at hX hY ⊢

Goal state after:
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f✝ f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y : LeftInvariantDerivation I G
g : G
f : C^∞⟮I, G; 𝕜⟯⟨g⟩
hX : (↑X ((Y f).comp (𝑳 I g))) 1 = ((Derivation.evalAt g) ↑X) (Y f)
hY : (↑Y ((X f).comp (𝑳 I g))) 1 = ((Derivation.evalAt g) ↑Y) (X f)
⊢ (⁅↑X, ↑Y⁆ (ContMDiffMap.comp f (𝑳 I g))) 1 = ((Derivation.evalAt g) ⁅↑X, ↑Y⁆) f
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

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
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → TopologicalSpace (V x)
inst✝¹ : FiberBundle F V
inst✝ : NormedSpace 𝕜 F
k : WithTop ℕ∞
x : M
σ₀ : V x
⊢ ContMDiffAt I (I.prod 𝓘(𝕜, F)) k (fun x' ↦ ⟨x', extend F σ₀ x'⟩) x

Tactic:
rw [contMDiffAt_section]

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
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
V : M → Type u_6
inst✝⁴ : TopologicalSpace (TotalSpace F V)
inst✝³ : (x : M) → AddCommGroup (V x)
inst✝² : (x : M) → TopologicalSpace (V x)
inst✝¹ : FiberBundle F V
inst✝ : NormedSpace 𝕜 F
k : WithTop ℕ∞
x : M
σ₀ : V x
⊢ ContMDiffAt I 𝓘(𝕜, F) k (fun x_1 ↦ (↑(trivializationAt F V x) ⟨x_1, extend F σ₀ x_1⟩).2) x
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
hV : V x = 0
hW : W x = 0
⊢ (mfderiv% ↑(extChartAt I x) x).inverse
      (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I))
        (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
        (↑(extChartAt I x) x)) =
    0

Tactic:
lieBracketWithin_eq_zero_of_eq_zero

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
hV : V x = 0
hW : W x = 0
⊢ (mfderiv% ↑(extChartAt I x) x).inverse 0 = 0

case hV
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
hV : V x = 0
hW : W x = 0
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I) (↑(extChartAt I x) x) = 0

case hW
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
hV : V x = 0
hW : W x = 0
⊢ mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I) (↑(extChartAt I x) x) = 0
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)
this :
  ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
    (fun x ↦
      fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm)
        ((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u) (↑(extChartAt I (g x₀)) (g x)))
    t' x₀
⊢ x₀ ∈ t'

Tactic:
simp [t', hx₀]

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

Goal state before:
pp: x y : ℝ
hxy : Fact (x < y)
z : EuclideanHalfSpace 1
hz : (↑z).ofLp 0 ≤ y - x
⊢ (↑z).ofLp 0 + x ∈ Icc x y

Tactic:
by simpa [z.prop, hxy.out.le, ← le_add_neg_iff_add_le]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
hf : ContinuousAt f x
equiv : (E × F) ≃L[𝕜] E''
domChart : OpenPartialHomeomorph M H
codChart : OpenPartialHomeomorph N G
hx : x ∈ domChart.source
hfx : f x ∈ codChart.source
hdomChart : domChart ∈ IsManifold.maximalAtlas I n M
hcodChart : codChart ∈ IsManifold.maximalAtlas J n N
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (domChart.extend I).target
⊢ IsImmersionAt I J n f x

Tactic:
by
  have aux : IsImmersionAtOfComplement F I J n f x := by
    apply IsImmersionAtOfComplement.mk_of_continuousAt <;> assumption
  use aux.smallComplement, by infer_instance, by infer_instance
  rwa [← IsImmersionAtOfComplement.congr_F aux.smallEquiv]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
w : E
hw : w ∈ (ℝ ∙ v)ᗮ
h₁ : 0 < ‖w‖ ^ 2 + 4
⊢ ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4

Tactic:
suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ ^ 2 = (‖w‖ ^ 2 + 4) ^ 2 by
    simpa only [sq_eq_sq_iff_abs_eq_abs, abs_norm, abs_of_pos h₁] using this

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
w : E
hw : w ∈ (ℝ ∙ v)ᗮ
h₁ : 0 < ‖w‖ ^ 2 + 4
⊢ ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ ^ 2 = (‖w‖ ^ 2 + 4) ^ 2
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(ℝ ∙ ↑(-v))ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) := ⋯
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (ℝ ∙ ↑(-v))ᗮ.subtypeL (U.symm 0)
⊢ (↑((ℝ ∙ ↑(-v))ᗮ.subtypeL ∘SL ↑↑U.symm)).range = (ℝ ∙ ↑v)ᗮ

Tactic:
convert!
    (U.symm : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] (ℝ ∙ (↑(-v) : E))ᗮ).range_comp
      (ℝ ∙ (↑(-v) : E))ᗮ.subtype using 1

Goal state after:
case e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(ℝ ∙ ↑(-v))ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) := (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subtype.val) (ℝ ∙ ↑(-v))ᗮ.subtypeL (U.symm 0)
⊢ (ℝ ∙ ↑v)ᗮ = (ℝ ∙ ↑(-v))ᗮ.subtype.range
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : BddAbove s
asup : ℝ := sSup s
hasup : asup = sSup s
a : ℝ
hlt : asup - ε / 2 < a
γ : ℝ → M
h0 : γ 0 = x
hγ : IsMIntegralCurveOn γ v (Ioo (-a) a)
γ1_aux : ℝ → M
h1_aux : γ1_aux 0 = γ (-(asup - ε / 2))
γ1 : ℝ → M := γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t + (asup - ε / 2) ∈ Ioo (-ε) ε}
γ1_def : γ1 = γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
heq1 : γ1 (-(asup - ε / 2)) = γ (-(asup - ε / 2))
γ2_aux : ℝ → M
h2_aux : γ2_aux 0 = γ (asup - ε / 2)
hγ2 : IsMIntegralCurveOn γ2_aux v (Ioo (-ε) ε)
⊢ False

Tactic:
rw [← isMIntegralCurveOn_comp_sub (dt := asup - ε / 2)] at hγ2

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
inst✝ : BoundarylessManifold I M
v : (x : M) → TangentSpace I x
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
ε : ℝ
hε : 0 < ε
h : ∀ (x : M), ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-ε) ε)
x : M
s : Set ℝ := {a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a)}
hbdd : BddAbove s
asup : ℝ := sSup s
hasup : asup = sSup s
a : ℝ
hlt : asup - ε / 2 < a
γ : ℝ → M
h0 : γ 0 = x
hγ : IsMIntegralCurveOn γ v (Ioo (-a) a)
γ1_aux : ℝ → M
h1_aux : γ1_aux 0 = γ (-(asup - ε / 2))
γ1 : ℝ → M := γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t + (asup - ε / 2) ∈ Ioo (-ε) ε}
γ1_def : γ1 = γ1_aux ∘ fun x ↦ x + (asup - ε / 2)
heq1 : γ1 (-(asup - ε / 2)) = γ (-(asup - ε / 2))
γ2_aux : ℝ → M
h2_aux : γ2_aux 0 = γ (asup - ε / 2)
hγ2 : IsMIntegralCurveOn (γ2_aux ∘ fun x ↦ x - (asup - ε / 2)) v {t | t - (asup - ε / 2) ∈ Ioo (-ε) ε}
⊢ False
Module: Mathlib.MeasureTheory.Constructions.UnitInterval

Goal state before:
pp: x : ↑I
⊢ volume (Ioi x) = ENNReal.ofReal (1 - ↑x)

Tactic:
simp only [volume_apply, image_subtype_val_Icc_Ioi, Real.volume_Ioc]

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b c : ℝ
γ : ℝ → M
h : a ≤ b
h' : b ≤ c
⊢ pathELength I γ a c = pathELength I γ a b + pathELength I γ b c

Tactic:
have : Icc a c = Icc a b ∪ Ioc b c := (Icc_union_Ioc_eq_Icc h h').symm

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b c : ℝ
γ : ℝ → M
h : a ≤ b
h' : b ≤ c
this : Icc a c = Icc a b ∪ Ioc b c
⊢ pathELength I γ a c = pathELength I γ a b + pathELength I γ b c
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
h : Fact (a < b)
γ : ↑(Icc a b) → M
⊢ ∫⁻ (t : ↑(Icc a b)), ‖(mfderiv% γ t) 1‖ₑ = pathELength I (γ ∘ projIcc a b ⋯) a b

Tactic:
rw [pathELength_eq_lintegral_mfderivWithin_Icc]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
h : Fact (a < b)
γ : ↑(Icc a b) → M
⊢ ∫⁻ (t : ↑(Icc a b)), ‖(mfderiv% γ t) 1‖ₑ = ∫⁻ (t : ℝ) in Icc a b, ‖(mfderiv[Icc a b] (γ ∘ projIcc a b ⋯) t) 1‖ₑ
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
n : ℕ∞ω
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
⊢ ContMDiffAt 𝓘(ℝ, F) (𝓘(ℝ, F).prod 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ)) ω (fun b ↦ ⟨b, innerSL ℝ⟩) x

Tactic:
rw [contMDiffAt_section]

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
n : ℕ∞ω
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
⊢ ContMDiffAt 𝓘(ℝ, F) 𝓘(ℝ, F →L[ℝ] F →L[ℝ] ℝ) ω
    (fun x_1 ↦
      (↑(trivializationAt (F →L[ℝ] F →L[ℝ] ℝ) (fun x ↦ TangentSpace 𝓘(ℝ, F) x →L[ℝ] TangentSpace 𝓘(ℝ, F) x →L[ℝ] ℝ) x)
          ⟨x_1, innerSL ℝ⟩).2)
    x
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
n : ℕ∞ω
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ∀ (i : Path x y), ContMDiff (𝓡∂ 1) 𝓘(ℝ, F) 1 ⇑i → edist x y ≤ ∫⁻ (x_1 : ↑unitInterval), ‖(mfderiv% ⇑i x_1) 1‖ₑ

Tactic:
intro γ hγ

Goal state after:
case refine_1
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
n : ℕ∞ω
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
γ : Path x y
hγ : ContMDiff (𝓡∂ 1) 𝓘(ℝ, F) 1 ⇑γ
⊢ edist x y ≤ ∫⁻ (x_1 : ↑unitInterval), ‖(mfderiv% ⇑γ x_1) 1‖ₑ