mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i j : J
inst✝ : IsCofilteredOrEmpty J
f : i ⟶ j
h : ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))
j' : J
f' : j' ⟶ j
k : J
g : k ⟶ i
g' : k ⟶ j'
he : g ≫ f = g' ≫ f'
⊢ range ⇑(ConcreteCategory.hom (F.map (g ≫ f))) ⊆ range ⇑(ConcreteCategory.hom (F.map f'))

Tactic:
he,

Goal state after:
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i j : J
inst✝ : IsCofilteredOrEmpty J
f : i ⟶ j
h : ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))
j' : J
f' : j' ⟶ j
k : J
g : k ⟶ i
g' : k ⟶ j'
he : g ≫ f = g' ≫ f'
⊢ range ⇑(ConcreteCategory.hom (F.map (g' ≫ f'))) ⊆ range ⇑(ConcreteCategory.hom (F.map f'))
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: case mp.refine_2
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
subi : {i} ⊆ {i, i'}
subi' : {i'} ⊆ {i, i'}
⊢ ↑(u (Opposite.op {i})) ⟨i, ⋯⟩ = ↑(u (Opposite.op {i'})) ⟨i', ⋯⟩ → i = i'

Tactic:
← hu (CategoryTheory.homOfLE subi).op,

Goal state after:
case mp.refine_2
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
i i' : ι
subi : {i} ⊆ {i, i'}
subi' : {i'} ⊆ {i, i'}
⊢ ↑((ConcreteCategory.hom ((hallMatchingsFunctor t).map (homOfLE subi).op)) (u (Opposite.op {i, i'}))) ⟨i, ⋯⟩ =
      ↑(u (Opposite.op {i'})) ⟨i', ⋯⟩ →
    i = i'
Module: Mathlib.Data.Matrix.PEquiv

Goal state before:
pp: k : Type u_1
l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝⁵ : Fintype n
inst✝⁴ : Fintype k
inst✝³ : DecidableEq n
inst✝² : DecidableEq k
inst✝¹ : DecidableEq m
inst✝ : Semiring α
a : m
b : n
c : k
M : Matrix k l α
⊢ (single a b).toMatrix * ((single b c).toMatrix * M) = (single a c).toMatrix * M

Tactic:
rw [← Matrix.mul_assoc, single_mul_single]

Goal state after:
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)
h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃)
h₁' : p ∈ line[R, t.points i₁, p₁]
h₂' : p ∈ line[R, t.points i₂, p₂]
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hp : p ∈ affineSpan R (Set.range t.points)
h₁i : p₁ ∈ ⇑(lineMap (t.points i₂) (t.points i₃)) '' Set.Ioo 0 1
h₂i : p₂ ∈ ⇑(lineMap (t.points i₁) (t.points i₃)) '' Set.Ioo 0 1
⊢ Sbtw R (t.points i₁) p p₁

Tactic:
Set.mem_image

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)
h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃)
h₁' : p ∈ line[R, t.points i₁, p₁]
h₂' : p ∈ line[R, t.points i₂, p₂]
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hp : p ∈ affineSpan R (Set.range t.points)
h₁i : ∃ x ∈ Set.Ioo 0 1, (lineMap (t.points i₂) (t.points i₃)) x = p₁
h₂i : ∃ x ∈ Set.Ioo 0 1, (lineMap (t.points i₁) (t.points i₃)) x = p₂
⊢ Sbtw R (t.points i₁) p p₁
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : ∃ M' ∈ doublyStochastic R n, M = s • M'
h✝ : IsEmpty n
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M

Tactic:
exact ⟨1, by simp, Subsingleton.elim _ _⟩

Goal state after:
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case refine_1.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x z : P
r : R
hr0 : 0 ≤ r
hr1 : r ≤ 1
hr0' : 0 < r
⊢ x = (lineMap x z) r ∨ z ∈ ⇑(lineMap x ((lineMap x z) r)) '' Set.Ici 1

Tactic:
Set.mem_image

Goal state after:
case refine_1.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x z : P
r : R
hr0 : 0 ≤ r
hr1 : r ≤ 1
hr0' : 0 < r
⊢ x = (lineMap x z) r ∨ ∃ x_1 ∈ Set.Ici 1, (lineMap x ((lineMap x z) r)) x_1 = z
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y z : P
h : Wbtw R x y z
⊢ x = y ∨ z ∈ ⇑(lineMap x y) '' Set.Ici 1

Tactic:
rcases h with ⟨r, ⟨hr0, hr1⟩, rfl⟩
    rcases hr0.lt_or_eq with (hr0' | rfl)
    · rw [Set.mem_image]
      refine .inr ⟨r⁻¹, (one_le_inv₀ hr0').2 hr1, ?_⟩
      simp only [lineMap_apply, smul_smul, vadd_vsub]
      rw [inv_mul_cancel₀ hr0'.ne', one_smul, vsub_vadd]
    · simp

Goal state after:
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: case refine_5
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∑ x ∈ t, w x • x = x

pp: case refine_5
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∀ x ∈ t, w x • x ≠ 0 → w x ≠ 0

Tactic:
· rw [t.centerMass_eq_of_sum_1 id hw₂] at hw₃
      exact hw₃

Goal state after:
case refine_5
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∀ x ∈ t, w x • x ≠ 0 → w x ≠ 0
Module: Mathlib.Analysis.Convex.Extrema

Goal state before:
pp: E : Type u_1
β : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module ℝ β
inst✝¹ : IsOrderedModule ℝ β
inst✝ : PosSMulReflectLE ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_localmin : IsLocalMinOn f s a
h_conv : ConvexOn ℝ s f
x : E
x_in_s : x ∈ s
g : ℝ →ᵃ[ℝ] E := AffineMap.lineMap a x
hg0 : g 0 = a
hg1 : g 1 = x
hgc : Continuous[_, inst✝⁹] ⇑g
h_maps : MapsTo (⇑g) (Icc 0 1) s
⊢ IsLocalMinOn (f ∘ ⇑g) (Icc 0 1) 0

Tactic:
← hg0

Goal state after:
E : Type u_1
β : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module ℝ β
inst✝¹ : IsOrderedModule ℝ β
inst✝ : PosSMulReflectLE ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_conv : ConvexOn ℝ s f
x : E
x_in_s : x ∈ s
g : ℝ →ᵃ[ℝ] E := AffineMap.lineMap a x
h_localmin : IsLocalMinOn f s (g 0)
hg0 : g 0 = a
hg1 : g 1 = x
hgc : Continuous[_, inst✝⁹] ⇑g
h_maps : MapsTo (⇑g) (Icc 0 1) s
⊢ IsLocalMinOn (f ∘ ⇑g) (Icc 0 1) 0
Module: Mathlib.Analysis.Convex.GaugeRescale

Goal state before:
pp: case inl
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hs : Convex ℝ s
hs₀ : s ∈ 𝓝 0
ht : Convex ℝ t
ht₀ : t ∈ 𝓝 0
htb : IsVonNBounded ℝ t
hta : Absorbent ℝ t
⊢ ContinuousAt (gaugeRescale s t) 0

Tactic:
rw [ContinuousAt, gaugeRescale_zero]
    nth_rewrite 2 [← comap_gauge_nhds_zero htb ht₀]
    simp only [tendsto_comap_iff, Function.comp_def, gauge_gaugeRescale _ hta htb]
    exact tendsto_gauge_nhds_zero hs₀

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
a : ℝ := ε / (ε + ‖x - y‖)
b : ℝ := ‖x - y‖ / (ε + ‖x - y‖)
hab : a + b = 1
hxyz : x = a • y + b • z
⊢ f x - f y ≤ ‖x - y‖ * (2 * M / ε)

Tactic:
← mul_div_assoc,

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
a : ℝ := ε / (ε + ‖x - y‖)
b : ℝ := ‖x - y‖ / (ε + ‖x - y‖)
hab : a + b = 1
hxyz : x = a • y + b • z
⊢ f x - f y ≤ ‖x - y‖ * (2 * M) / ε
Module: Mathlib.Analysis.Convex.Intrinsic

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁸ : Ring 𝕜
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module 𝕜 V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module 𝕜 W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f : P → Q
s : Set P
e : ↥(affineSpan 𝕜 s) → ↥(affineSpan 𝕜 (f '' s))
he_homeo : IsHomeomorph e
he : ∀ (x : ↥(affineSpan 𝕜 s)), ↑(e x) = f ↑x
x : ↥(affineSpan 𝕜 s)
x✝ : x ∈ f ∘ Subtype.val ⁻¹' f '' s
w✝ : P
hy : w✝ ∈ s
hfy : f w✝ = (f ∘ Subtype.val) x
⊢ ↑(e x) = ↑(e ⟨w✝, ⋯⟩)

Tactic:
by simpa [he] using hfy.symm

Goal state after:
Module: Mathlib.Analysis.Convex.Intrinsic

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁸ : Ring 𝕜
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module 𝕜 V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module 𝕜 W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f : P → Q
s : Set P
e : [Nonempty ↑s] → ↥(affineSpan 𝕜 s) → ↥(affineSpan 𝕜 (f '' s))
he_homeo : ∀ [inst : Nonempty ↑s], IsHomeomorph e
he : ∀ [inst : Nonempty ↑s] (x : ↥(affineSpan 𝕜 s)), ↑(e x) = f ↑x
⊢ intrinsicClosure 𝕜 (f '' s) = f '' intrinsicClosure 𝕜 s

Tactic:
by
  rcases s.eq_empty_or_nonempty with rfl | hs
  · simp
  · have : Nonempty s := hs.to_subtype
    rw [intrinsicClosure, ← image_closure_preimage_comp e he_homeo,
      (funext he : (↑) ∘ e = f ∘ (↑)),
      preimage_image_eq_of_homeomorph_affineSpan e he_homeo he, image_comp]; rfl

Goal state after:
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
g : E → ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConvexOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
this : ∀ {t : Set α}, μ t ≠ 0 → ⨍ (x : α) in t, f x ∂μ ∈ s ∧ g (⨍ (x : α) in t, f x ∂μ) ≤ ⨍ (x : α) in t, g (f x) ∂μ
t : Set α
hm : MeasurableSet t
h₀ : μ t ≠ 0
h₀' : μ tᶜ ≠ 0
hne : ⨍ (x : α) in t, f x ∂μ ≠ ⨍ (x : α) in tᶜ, f x ∂μ
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
h_avg :
  a • ⨍ (x : α) in t, f x ∂μ + b • ⨍ (x : α) in tᶜ, f x ∂μ = ⨍ (x : α), f x ∂μ ∧
    a • ⨍ (x : α) in t, g (f x) ∂μ + b • ⨍ (x : α) in tᶜ, g (f x) ∂μ = ⨍ (x : α), g (f x) ∂μ
⊢ g (a • ⨍ (x : α) in t, f x ∂μ + b • ⨍ (x : α) in tᶜ, f x ∂μ) <
    a • ⨍ (x : α) in t, g (f x) ∂μ + b • ⨍ (x : α) in tᶜ, g (f x) ∂μ

Tactic:
calc
    g ((a • ⨍ x in t, f x ∂μ) + b • ⨍ x in tᶜ, f x ∂μ) <
        a * g (⨍ x in t, f x ∂μ) + b * g (⨍ x in tᶜ, f x ∂μ) :=
      hg.2 (this h₀).1 (this h₀').1 hne ha hb hab
    _ ≤ (a * ⨍ x in t, g (f x) ∂μ) + b * ⨍ x in tᶜ, g (f x) ∂μ := by
      gcongr
      exacts [(this h₀).2, (this h₀').2]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y : P
⊢ ((Finsupp.single x s + Finsupp.single y t).sum fun x r ↦ r) = 1

Tactic:
· rw [Finsupp.sum_add_index (by simp) (by simp), Finsupp.sum_single_index (by simp),
        Finsupp.sum_single_index (by simp), h]

Goal state after:
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: case inr.hst
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : ι → Set E
h_convex : ∀ (i : ι), Convex 𝕜 (F i)
h_compact : ∀ (i : ι), IsCompact (F i)
h_inter : ∀ (I : Finset ι), #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
h_nonempty : Nonempty ι
h_fin : ∀ (I : Finset ι), (⋂ i ∈ I, F i).Nonempty
i0 : ι
I : Finset ι
⊢ (F i0 ∩ ⋂ i ∈ I, F i).Nonempty

Tactic:
simpa using h_fin ({i0} ∪ I)

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
h : s.WSameSide x y
f : P →ᵃ[R] P'
⊢ (AffineSubspace.map f s).WSameSide (f x) (f y)

Tactic:
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
  refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
  simp_rw [← linearMap_vsub]
  exact h.map f.linear

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : PartialOrder R
inst✝⁶ : IsStrictOrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
s : AffineSubspace R P
x y : P
h : s.WOppSide x y
f : P →ᵃ[R] P'
⊢ (AffineSubspace.map f s).WOppSide (f x) (f y)

Tactic:
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
  refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
  simp_rw [← linearMap_vsub]
  exact h.map f.linear

Goal state after:
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case inr.inr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p
h : p₂ -ᵥ p ∈ (fun θ ↦ (1 - θ) • (p₁ -ᵥ p) + θ • (p₃ -ᵥ p)) '' Set.Ioo 0 1
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)

Tactic:
Set.mem_image

Goal state after:
case inr.inr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p
h : ∃ x ∈ Set.Ioo 0 1, (1 - x) • (p₁ -ᵥ p) + x • (p₃ -ᵥ p) = p₂ -ᵥ p
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₂ : P
h : p₂ ∈ s
⊢ s.SSameSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
by
  rw [sSameSide_comm, sSameSide_iff_exists_left h, ← and_assoc, and_comm (a := y ∉ s), and_assoc]
  simp_rw [SameRay.sameRay_comm]

Goal state after:
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case inr
E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
r : ℝ
x y z : PE
hxy : dist x y = r * dist x z
hyz : dist y z = (1 - r) * dist x z
this : y -ᵥ x ∈ [0 -[ℝ] z -ᵥ x]
hne : x ≠ z
⊢ y = (AffineMap.lineMap x z) r

Tactic:
rw [← dist_ne_zero] at hne
    obtain ⟨a, b, _, hb, _, H⟩ := this
    rw [smul_zero, zero_add] at H
    have H' := congr_arg norm H
    rw [norm_smul, Real.norm_of_nonneg hb, ← dist_eq_norm_vsub', ← dist_eq_norm_vsub', hxy,
      mul_left_inj' hne] at H'
    rw [AffineMap.lineMap_apply, ← H', H, vsub_vadd]

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide ((Finset.affineCombination R Finset.univ s.points) w₁)
      ((Finset.affineCombination R Finset.univ s.points) w₂) ↔
    SignType.sign (w₁ i) = SignType.sign (w₂ i) ∨ w₁ i = 0 ∨ w₂ i = 0

Tactic:
by
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · by_cases h0 : w₁ i = 0
    · simp [h0]
    by_cases h0' : w₂ i = 0
    · simp [h0']
    exact .inl ((s.sSameSide_affineSpan_faceOpposite_iff hw₁ hw₂).1 ⟨h,
      (s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).not.2 h0,
      (s.affineCombination_mem_affineSpan_faceOpposite_iff hw₂).not.2 h0'⟩).1
  · by_cases h0 : w₁ i = 0
    · exact wSameSide_of_left_mem _ ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).2 h0)
    · by_cases h0' : w₂ i = 0
      · exact wSameSide_of_right_mem _
          ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₂).2 h0')
      simp only [h0, h0', or_self, or_false] at h
      exact (s.sSameSide_affineSpan_faceOpposite_of_sign_eq hw₁ hw₂ h h0).wSameSide

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case succ
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
h'f : HasTemperateGrowth (fderiv ℝ f)
hf : Differentiable ℝ f
k : ℕ
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k
m : ℕ
⊢ ∃ k C, ∀ (x : E), ‖iteratedFDeriv ℝ (m + 1) f x‖ ≤ C * (1 + ‖x‖) ^ k

Tactic:
rcases h'f.2 m with ⟨k', C', h'⟩

Goal state after:
case succ
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
h'f : HasTemperateGrowth (fderiv ℝ f)
hf : Differentiable ℝ f
k : ℕ
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k
m k' : ℕ
C' : ℝ
h' : ∀ (x : E), ‖iteratedFDeriv ℝ m (fderiv ℝ f) x‖ ≤ C' * (1 + ‖x‖) ^ k'
⊢ ∃ k C, ∀ (x : E), ‖iteratedFDeriv ℝ (m + 1) f x‖ ≤ C * (1 + ‖x‖) ^ k
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (g t)

Tactic:
rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (g t)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E →L[ℝ] F
⊢ HasTemperateGrowth (fderiv ℝ ⇑f)

Tactic:
have : fderiv ℝ f = fun _ ↦ f := by ext1 v; simp only [ContinuousLinearMap.fderiv]
    simp [this]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
⊢ T x₀ = T.rayleighQuotient x₀ • x₀

Tactic:
obtain ⟨a, b, h₁, h₂⟩ := hT.linearly_dependent_of_isLocalExtrOn hextr

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
T : F →L[ℝ] F
hT : IsSelfAdjoint T
x₀ : F
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
a b : ℝ
h₁ : (a, b) ≠ 0
h₂ : a • x₀ + b • T x₀ = 0
⊢ T x₀ = T.rayleighQuotient x₀ • x₀
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
⊢ (f ∘ ⇑σ) ∘ ⇑(sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)

Tactic:
rw [Function.comp_assoc, ← Equiv.Perm.coe_mul]

Goal state after:
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
⊢ f ∘ ⇑(σ * sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)
Module: Mathlib.Analysis.Matrix.Hermitian

Goal state before:
pp: n : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : E →ₗ[𝕜] E
b : OrthonormalBasis n 𝕜 E
⊢ ((toMatrix b.toBasis b.toBasis) f).IsHermitian ↔ f.IsSymmetric

Tactic:
by
  rw [← Matrix.isSymmetric_toLin_iff b, Matrix.toLin_toMatrix]

Goal state after: