mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)
L : F₁ →SL[σ] F₂
v : F₁
⊢ (Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (L ((Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) =
    (Trivialization.continuousLinearMapAt 𝕜₂ e₂' b)
      ((Trivialization.symmL 𝕜₂ e₂ b)
        (L ((Trivialization.continuousLinearMapAt 𝕜₁ e₁ b) ((Trivialization.symmL 𝕜₁ e₁' b) v))))

Tactic:
Trivialization.continuousLinearMapAt_apply

Goal state after:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)
L : F₁ →SL[σ] F₂
v : F₁
⊢ (Trivialization.coordChangeL 𝕜₂ e₂ e₂' b) (L ((Trivialization.coordChangeL 𝕜₁ e₁' e₁ b) v)) =
    (Trivialization.linearMapAt 𝕜₂ e₂' b)
      ((Trivialization.symmL 𝕜₂ e₂ b) (L ((Trivialization.linearMapAt 𝕜₁ e₁ b) ((Trivialization.symmL 𝕜₁ e₁' b) v))))
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W : (x : E) → TangentSpace 𝓘(𝕜, E) x
s : Set E
⊢ mlieBracketWithin 𝓘(𝕜, E) V W s = lieBracketWithin 𝕜 V W s

Tactic:
by
  ext x
  simp [mlieBracketWithin_apply]

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

Goal state before:
pp: case e'_11
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
hg' :
  ContinuousWithinAt g t' x₀ ∧
    ContDiffWithinAt 𝕜 m (↑(extChartAt I (g x₀)) ∘ g ∘ ↑(extChartAt J x₀).symm)
      (↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)
hf' :
  ContinuousWithinAt (uncurry f) (t' ×ˢ u) (x₀, g x₀) ∧
    ContDiffWithinAt 𝕜 n
      (fun x ↦ ↑(extChartAt I' (f x₀ (g x₀))) (f (↑(extChartAt J x₀).symm x.1) (↑(extChartAt I (g x₀)).symm x.2)))
      ((fun p ↦ (↑(extChartAt J x₀).symm p.1, ↑(extChartAt I (g x₀)).symm p.2)) ⁻¹' t' ×ˢ u ∩ range ↑J ×ˢ range ↑I)
      (↑((extChartAt J x₀).prod (extChartAt I (g x₀))) (x₀, g x₀))
this :
  (fun p ↦ (↑(extChartAt J x₀).symm p.1, ↑(extChartAt I (g x₀)).symm p.2)) ⁻¹' t' ×ˢ u ∩
      range ↑J ×ˢ (extChartAt I (g x₀)).target ⊆
    (fun p ↦ (↑(extChartAt J x₀).symm p.1, ↑(extChartAt I (g x₀)).symm p.2)) ⁻¹' t' ×ˢ u ∩ range ↑J ×ˢ range ↑I
y : F × E
⊢ (↑(chartAt G x₀).symm (↑J.symm y.1) ∈ t' ∧ ∃ y_1, ↑J y_1 = y.1) ∧
      ((∃ y_1, ↑I y_1 = y.2) ∧ ↑I.symm y.2 ∈ (chartAt H (g x₀)).target) ∧ ↑(chartAt H (g x₀)).symm (↑I.symm y.2) ∈ u ↔
    (↑(chartAt G x₀).symm (↑J.symm y.1) ∈ t' ∧ ↑(chartAt H (g x₀)).symm (↑I.symm y.2) ∈ u) ∧
      (∃ y_1, ↑J y_1 = y.1) ∧ (∃ y_1, ↑I y_1 = y.2) ∧ ↑I.symm y.2 ∈ (chartAt H (g x₀)).target

Tactic:
tauto

Goal state after:
Module: Mathlib.Topology.Algebra.Module.TransferInstance

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : Group β
inst✝² : IsTopologicalGroup β
inst✝¹ : TopologicalSpace α
inst✝ : Group α
e : α ≃ₜ* β
f : β × β → β := fun q ↦ q.1 * q.2
⊢ Continuous fun p ↦ p.1 * p.2

Tactic:
have : Continuous (fun p ↦ e.symm <| f (e p.1, e p.2) : (α × α → α)) := by fun_prop

Goal state after:
α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : Group β
inst✝² : IsTopologicalGroup β
inst✝¹ : TopologicalSpace α
inst✝ : Group α
e : α ≃ₜ* β
f : β × β → β := fun q ↦ q.1 * q.2
this : Continuous fun p ↦ e.symm (f (e p.1, e p.2))
⊢ Continuous fun p ↦ p.1 * p.2
Module: Mathlib.Geometry.Manifold.Instances.Real

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

Tactic:
linarith

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

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
z : EuclideanSpace ℝ (Fin 1)
hz : 0 ≤ z.ofLp 0
h'z : z.ofLp 0 < y - x
i : Fin 1
⊢ x + z.ofLp 0 ≤ y

Tactic:
linarith

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 ≤ y

Tactic:
linarith

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

Goal state before:
pp: x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : EuclideanHalfSpace 1
hz : (↑z).ofLp 0 < y - x
⊢ x < y - (↑z).ofLp 0

Tactic:
linarith

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

Goal state before:
pp: x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄, x_1 ∈ {z | (↑z).ofLp 0 < y - x} → ⟨max (y - (↑x_1).ofLp 0) x, ⋯⟩ ∈ {z | x < ↑z}

Tactic:
simp only [lt_max_iff, mem_ofPred_eq]; intro z hz; left
    linarith

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

Goal state before:
pp: x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : EuclideanSpace ℝ (Fin 1)
hz : 0 ≤ z.ofLp 0
h'z : z.ofLp 0 < y - x
i : Fin 1
⊢ x ≤ y - z.ofLp 0

Tactic:
linarith

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

Goal state before:
pp: x✝ y✝ : ℝ
hxy : Fact (x✝ < y✝)
x y : ℝ
h : Fact (x < y)
z : EuclideanSpace ℝ (Fin 1)
hz : 0 ≤ z.ofLp 0
h'z : z.ofLp 0 < y - x
i : Fin 1
A : x ≤ y - z.ofLp 0
⊢ (toLp 2 fun x_1 ↦ y - ↑⟨max (y - (↑⟨z, hz⟩).ofLp 0) x, ⋯⟩).ofLp 0 = z.ofLp 0

Tactic:
simp only [Fin.isValue, A, sup_of_le_left, sub_sub_cancel]

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
⊢ x ≤ y - (↑z).ofLp 0

Tactic:
linarith

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

Goal state before:
pp: case h
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
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
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (domChart.extend I).target
aux : IsImmersionAtOfComplement F I J n f x
⊢ IsImmersionAtOfComplement aux.smallComplement I J n f x

Tactic:
← IsImmersionAtOfComplement.congr_F aux.smallEquiv

Goal state after:
case h
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
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
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (domChart.extend I).target
aux : IsImmersionAtOfComplement F I J n f x
⊢ IsImmersionAtOfComplement F I J n f x
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: case h
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
aux : IsImmersionAtOfComplement F I J n f x
⊢ IsImmersionAtOfComplement aux.smallComplement I J n f x

Tactic:
← IsImmersionAtOfComplement.congr_F aux.smallEquiv

Goal state after:
case h
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
aux : IsImmersionAtOfComplement F I J n f x
⊢ IsImmersionAtOfComplement F I J n f x
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
u : EuclideanHalfSpace 1
hz' : ↑(𝓡∂ 1) u ∈ ((chartAt (EuclideanHalfSpace 1) z).extend (𝓡∂ 1)).target
hu : (↑u).ofLp 0 ≤ y - x
⊢ y / 2 - (y - (↑u).ofLp 0) + y / 2 = (↑u).ofLp 0

Tactic:
linarith

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 : ↥(ℝ ∙ v)ᗮ
hw : ⟪v, ↑w⟫_ℝ = 0
⊢ ‖↑w‖ ^ 2 - 4 < 1 * (‖↑w‖ ^ 2 + 4)

Tactic:
linarith

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
x : ↑(sphere 0 1)
hx : ↑x ≠ v
a : ℝ := ((innerSL ℝ) v) ↑x
y : ↥(ℝ ∙ v)ᗮ := (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x
⊢ (‖(2 / (1 - a)) • y‖ ^ 2 + 4)⁻¹ • 4 • ↑((2 / (1 - a)) • y) +
      (‖(2 / (1 - a)) • y‖ ^ 2 + 4)⁻¹ • (‖(2 / (1 - a)) • y‖ ^ 2 - 4) • v =
    ↑x

Tactic:
have split : ↑x = a • v + ↑y := by
    rw [← ((ℝ ∙ v).starProjection_add_starProjection_orthogonal x),
      Submodule.starProjection_unit_singleton ℝ hv x]
    rfl

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
x : ↑(sphere 0 1)
hx : ↑x ≠ v
a : ℝ := ((innerSL ℝ) v) ↑x
y : ↥(ℝ ∙ v)ᗮ := (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x
split : ↑x = a • v + ↑y
⊢ (‖(2 / (1 - a)) • y‖ ^ 2 + 4)⁻¹ • 4 • ↑((2 / (1 - a)) • y) +
      (‖(2 / (1 - a)) • y‖ ^ 2 + 4)⁻¹ • (‖(2 / (1 - a)) • y‖ ^ 2 - 4) • v =
    ↑x
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
x : ↑(sphere 0 1)
hx : ↑x ≠ v
a : ℝ := ((innerSL ℝ) v) ↑x
y : ↥(ℝ ∙ v)ᗮ := (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x
split : ↑x = a • v + ↑y
hvy : ⟪v, ↑y⟫_ℝ = 0
pythag : 1 = a ^ 2 + ‖y‖ ^ 2
this : a < 1
⊢ 0 < 1 - a

Tactic:
linarith

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: case ha
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
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ -a ≤ -a'

Tactic:
linarith

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: case hb
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
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ a' ≤ a

Tactic:
linarith

Goal state after:
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
⊢ ∃ γ, γ 0 = x ∧ IsMIntegralCurve γ v

Tactic:
let s := { a | ∃ γ, γ 0 = x ∧ IsMIntegralCurveOn γ v (Ioo (-a) a) }

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)}
⊢ ∃ γ, γ 0 = x ∧ IsMIntegralCurve γ v
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)
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)
γ2 : ℝ → M := γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
γ2_def : γ2 = γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t ∈ Ioo (-ε - (asup - ε / 2)) (ε - (asup - ε / 2))}
hγ2 : IsMIntegralCurveOn γ2 v {t | -ε < t - (asup - ε / 2) ∧ t - (asup - ε / 2) < ε}
⊢ False

Tactic:
lt_sub_iff_add_lt,

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)
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)
γ2 : ℝ → M := γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
γ2_def : γ2 = γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t ∈ Ioo (-ε - (asup - ε / 2)) (ε - (asup - ε / 2))}
hγ2 : IsMIntegralCurveOn γ2 v {t | -ε + (asup - ε / 2) < t ∧ t - (asup - ε / 2) < ε}
⊢ False
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)
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)
γ2 : ℝ → M := γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
γ2_def : γ2 = γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t ∈ Ioo (-ε - (asup - ε / 2)) (ε - (asup - ε / 2))}
hγ2 : IsMIntegralCurveOn γ2 v {t | t ∈ Ioo (-ε + (asup - ε / 2)) (ε + (asup - ε / 2))}
hεle : ε ≤ asup
γ_ext : ℝ → M := (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
γ_ext_def : γ_ext = (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
⊢ -(asup + ε / 2) < 0

Tactic:
linarith

Goal state after:
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)
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)
γ2 : ℝ → M := γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
γ2_def : γ2 = γ2_aux ∘ fun x ↦ x - (asup - ε / 2)
heq2 : γ2 (asup - ε / 2) = γ (asup - ε / 2)
hγ1 : IsMIntegralCurveOn γ1 v {t | t ∈ Ioo (-ε - (asup - ε / 2)) (ε - (asup - ε / 2))}
hγ2 : IsMIntegralCurveOn γ2 v {t | t ∈ Ioo (-ε + (asup - ε / 2)) (ε + (asup - ε / 2))}
hεle : ε ≤ asup
γ_ext : ℝ → M := (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
γ_ext_def : γ_ext = (Ioo (-(asup + ε / 2)) a).piecewise ((Ioo (-a) a).piecewise γ γ1) γ2
⊢ 0 < a

Tactic:
linarith

Goal state after: