mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem

Goal state before:
pp: case h₂f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a₀ : E
f : ℂ → E
r : ℝ
h : Meromorphic f
h₁f : CircleIntegrable (fun x ↦ log⁺ ‖f x‖) 0 r
h₂f : CircleIntegrable (fun x ↦ log⁺ ‖f x - a₀‖) 0 r
⊢ ∀ x ∈ sphere 0 |r|, |(fun x ↦ log⁺ ‖f x‖) - fun x ↦ log⁺ ‖f x - a₀‖| x ≤ log⁺ ‖a₀‖ + log 2

Tactic:
intro θ

Goal state after:
case h₂f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a₀ : E
f : ℂ → E
r : ℝ
h : Meromorphic f
h₁f : CircleIntegrable (fun x ↦ log⁺ ‖f x‖) 0 r
h₂f : CircleIntegrable (fun x ↦ log⁺ ‖f x - a₀‖) 0 r
θ : ℂ
⊢ θ ∈ sphere 0 |r| → |(fun x ↦ log⁺ ‖f x‖) - fun x ↦ log⁺ ‖f x - a₀‖| θ ≤ log⁺ ‖a₀‖ + log 2
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: case inr.hs
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
h : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))
x : ℝ
xs : x ∈ s
y : ℝ
ys : y ∈ s
yx : y ≤ x
⊢ (s ∩ Icc x y).Subsingleton

Tactic:
rintro z ⟨zs, xz, zy⟩ w ⟨ws, xw, wy⟩
      cases le_antisymm (zy.trans yx) xz
      cases le_antisymm (wy.trans yx) xw
      rfl

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : DecidableEq E
inst✝ : ProperSpace E
e : E
⊢ 1 =o[atTop] logCounting (single e 1)

Tactic:
have hΘ : (fun r ↦ log r - log ‖e‖) =Θ[atTop] log :=
    (IsEquivalent.sub_isLittleO IsEquivalent.refl isLittleO_const_log_atTop).isTheta

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : DecidableEq E
inst✝ : ProperSpace E
e : E
hΘ : (fun r ↦ log r - log ‖e‖) =Θ[atTop] log
⊢ 1 =o[atTop] logCounting (single e 1)
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
e : WithTop E
r : ℝ
h : e = ⊤
⊢ logCounting f e (-r) = logCounting f e r

Tactic:
simp [logCounting, h, locallyFinsuppWithin.logCounting_even _ r]

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : ProperSpace 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
e : WithTop E
r : ℝ
h : ¬e = ⊤
⊢ logCounting f e (-r) = logCounting f e r

Tactic:
simp [logCounting, h, locallyFinsuppWithin.logCounting_even _ r]

Goal state after:
Module: Mathlib.Combinatorics.Hall.Finite

Goal state before:

Tactic:
Fintype.card s = #s := Fintype.card_coe _
      _ < Fintype.card ι := (card_lt_iff_ne_univ _).mpr hns
      _ = n.succ := hn

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

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✝⁷ : Ring R
inst✝⁶ : PartialOrder 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'
x y z : P
f : P ≃ᵃ[R] P'
⊢ Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z

Tactic:
have : Function.Injective f.toAffineMap := f.injective
  -- `refine` or `exact` are very slow, `apply` is fast. Please check before golfing.
  apply this.sbtw_map_iff

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

Goal state before:
pp: case pos
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 : AddTorsor V P
inst✝ : Nontrivial P
p₁ : P
h : List.Sbtw R [p₁]
hl : ¬[p₁] = []
⊢ ∃ p₁_1 p₂, p₁_1 ≠ p₂ ∧ ∃ l', l'.SortedLT ∧ map (⇑(lineMap p₁_1 p₂)) l' = [p₁]

Tactic:
rcases exists_ne p₁ with ⟨p₂, hp₂p₁⟩

Goal state after:
case pos
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 : AddTorsor V P
inst✝ : Nontrivial P
p₁ : P
h : List.Sbtw R [p₁]
hl : ¬[p₁] = []
p₂ : P
hp₂p₁ : p₂ ≠ p₁
⊢ ∃ p₁_1 p₂, p₁_1 ≠ p₂ ∧ ∃ l', l'.SortedLT ∧ map (⇑(lineMap p₁_1 p₂)) l' = [p₁]
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
w x y z : P
h₁ : Wbtw R w x z
h₂ : Wbtw R x y z
h : w ≠ x
⊢ w ≠ y

Tactic:
by
  rintro rfl
  exact h (h₁.swap_left_iff.1 h₂)

Goal state after:
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case refine_1
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
h : F.IsMittagLeffler
j j₁ : J
g₁ : j₁ ⟶ i
f₁ : j₁ ⟶ j
j₂ : J
f₂ : j₂ ⟶ j₁
h₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))
j₃ : J
f₃ : j₃ ⟶ j₂
x : F.obj j₂
hx : x ∈ ⋂ f, ⇑(ConcreteCategory.hom (F.map f)) ⁻¹' s
this : (ConcreteCategory.hom (F.map f₂)) x ∈ F.eventualRange j₁
g₂ : j₃ ⟶ i
j₄ : J
f₄ : j₄ ⟶ j₃
h₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁
y : F.obj j₄
hy : (ConcreteCategory.hom (F.map f₄)) y ∈ F.eventualRange j₃
h₃ : (ConcreteCategory.hom (F.map (f₄ ≫ f₃ ≫ f₂))) y = (ConcreteCategory.hom (F.map f₂)) x
⊢ (ConcreteCategory.hom (F.map g₁)) ((ConcreteCategory.hom (F.map (f₄ ≫ f₃ ≫ f₂))) y) ∈ s

Tactic:
h₃,

Goal state after:
case refine_1
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
h : F.IsMittagLeffler
j j₁ : J
g₁ : j₁ ⟶ i
f₁ : j₁ ⟶ j
j₂ : J
f₂ : j₂ ⟶ j₁
h₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))
j₃ : J
f₃ : j₃ ⟶ j₂
x : F.obj j₂
hx : x ∈ ⋂ f, ⇑(ConcreteCategory.hom (F.map f)) ⁻¹' s
this : (ConcreteCategory.hom (F.map f₂)) x ∈ F.eventualRange j₁
g₂ : j₃ ⟶ i
j₄ : J
f₄ : j₄ ⟶ j₃
h₄ : f₄ ≫ g₂ = f₄ ≫ (f₃ ≫ f₂) ≫ g₁
y : F.obj j₄
hy : (ConcreteCategory.hom (F.map f₄)) y ∈ F.eventualRange j₃
h₃ : (ConcreteCategory.hom (F.map (f₄ ≫ f₃ ≫ f₂))) y = (ConcreteCategory.hom (F.map f₂)) x
⊢ (ConcreteCategory.hom (F.map g₁)) ((ConcreteCategory.hom (F.map f₂)) x) ∈ s
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case refine_2
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
h : F.IsMittagLeffler
j j₁ : J
g₁ : j₁ ⟶ i
f₁ : j₁ ⟶ j
j₂ : J
f₂ : j₂ ⟶ j₁
h₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))
j₃ : J
f₃ : j₃ ⟶ j₂
x : F.obj j₂
hx : x ∈ ⋂ f, ⇑(ConcreteCategory.hom (F.map f)) ⁻¹' s
this : (ConcreteCategory.hom (F.map f₂)) x ∈ F.eventualRange j₁
y : F.obj j₃
hy : y ∈ F.eventualRange j₃
h₃ : (ConcreteCategory.hom (F.map (f₃ ≫ f₂))) y = (ConcreteCategory.hom (F.map f₂)) x
⊢ (ConcreteCategory.hom (F.map f₁)) ((ConcreteCategory.hom (F.map (f₃ ≫ f₂))) y) =
    (ConcreteCategory.hom (F.map (f₂ ≫ f₁))) x

Tactic:
h₃,

Goal state after:
case refine_2
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
i : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
h : F.IsMittagLeffler
j j₁ : J
g₁ : j₁ ⟶ i
f₁ : j₁ ⟶ j
j₂ : J
f₂ : j₂ ⟶ j₁
h₂ : F.eventualRange j₁ = range ⇑(ConcreteCategory.hom (F.map f₂))
j₃ : J
f₃ : j₃ ⟶ j₂
x : F.obj j₂
hx : x ∈ ⋂ f, ⇑(ConcreteCategory.hom (F.map f)) ⁻¹' s
this : (ConcreteCategory.hom (F.map f₂)) x ∈ F.eventualRange j₁
y : F.obj j₃
hy : y ∈ F.eventualRange j₃
h₃ : (ConcreteCategory.hom (F.map (f₃ ≫ f₂))) y = (ConcreteCategory.hom (F.map f₂)) x
⊢ (ConcreteCategory.hom (F.map f₁)) ((ConcreteCategory.hom (F.map f₂)) x) = (ConcreteCategory.hom (F.map (f₂ ≫ f₁))) x
Module: Mathlib.LinearAlgebra.Matrix.Permutation

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝² : DecidableEq n
σ τ : Perm n
inst✝¹ : Fintype n
inst✝ : NonAssocSemiring R
⊢ Perm.permMatrix R (σ * τ) = Perm.permMatrix R τ * Perm.permMatrix R σ

Tactic:
by
  rw [Perm.permMatrix, Perm.mul_def, toPEquiv_trans, PEquiv.toMatrix_trans]

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}
hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points)
⊢ p₁ ∈ affineSpan R (Set.range t.points)

Tactic:
rw [AffineSubspace.le_def'] at hle

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}
hle : ∀ p ∈ line[R, t.points i₂, t.points i₃], p ∈ affineSpan R (Set.range t.points)
⊢ p₁ ∈ affineSpan R (Set.range t.points)
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}
hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points)
⊢ p ∈ affineSpan R (Set.range t.points)

Tactic:
rw [AffineSubspace.le_def'] at hle

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}
hle : ∀ p ∈ line[R, t.points i₁, p₁], p ∈ affineSpan R (Set.range t.points)
⊢ p ∈ affineSpan R (Set.range t.points)
Module: Mathlib.LinearAlgebra.Matrix.Stochastic

Goal state before:
pp: R✝ : Type u_1
n✝ : Type u_2
inst✝⁹ : Fintype n✝
inst✝⁸ : DecidableEq n✝
inst✝⁷ : Semiring R✝
inst✝⁶ : PartialOrder R✝
inst✝⁵ : IsOrderedRing R✝
M✝ : Matrix n✝ n✝ R✝
x : n✝ → R✝
R : Type u_3
n : Type u_4
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
M N : Matrix n n R
hM : M ∈ {M | (∀ (i j : n), 0 ≤ M i j) ∧ 1 ᵥ* M = 1}
hN : N ∈ {M | (∀ (i j : n), 0 ≤ M i j) ∧ 1 ᵥ* M = 1}
⊢ M * N ∈ {M | (∀ (i j : n), 0 ≤ M i j) ∧ 1 ᵥ* M = 1}

Tactic:
refine Set.mem_sep ?_ ?_
    · intro i j
      apply Finset.sum_nonneg
      grind [mul_nonneg]
    · rw [← vecMul_vecMul, hM.2, hN.2]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Stochastic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix n n R
e₁ e₂ : n ≃ m
⊢ (reindex e₁ e₂) M ∈ rowStochastic R m ↔ M ∈ rowStochastic R n

Tactic:
refine ⟨fun h => ?_, reindex_mem_rowStochastic⟩

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix n n R
e₁ e₂ : n ≃ m
h : (reindex e₁ e₂) M ∈ rowStochastic R m
⊢ M ∈ rowStochastic R n
Module: Mathlib.Analysis.Convex.Body

Goal state before:
pp: V : Type u_1
inst✝¹ : SeminormedAddCommGroup V
inst✝ : NormedSpace ℝ V
K L : ConvexBody V
⊢ Metric.hausdorffEDist ↑K ↑L = edist K L

Tactic:
by
  rw [edist_dist]
  exact (ENNReal.ofReal_toReal hausdorffEDist_ne_top).symm

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Dual

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : TopologicalSpace F
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module ℝ E
inst✝³ : ContinuousSMul ℝ E
inst✝² : LocallyConvexSpace ℝ E
inst✝¹ : Module ℝ F
p : E →ₗ[ℝ] F →ₗ[ℝ] ℝ
inst✝ : p.IsContPerfPair
C : ProperCone ℝ E
⊢ dual p.flip ↑(dual p ↑C) = C

Tactic:
refine le_antisymm (fun x ↦ ?_) subset_dual_dual
  simp only [mem_dual, SetLike.mem_coe]
  contrapose!
  simpa [p.flip.toContPerfPair.surjective.exists] using C.hyperplane_separation_point

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.TensorProduct

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : Module R G
H : Type u_3
inst✝¹ : AddCommGroup H
inst✝ : Module R H
C₁ : PointedCone R G
C₂ : PointedCone R H
z : H ⊗[R] G
hz : ∀ φ ∈ dual (Dual.eval R H) ↑C₂, ∀ ψ ∈ dual (Dual.eval R G) ↑C₁, 0 ≤ ((dualDistrib R H G) (φ ⊗ₜ[R] ψ)) z
φ : Dual R G
hφ : φ ∈ dual (Dual.eval R G) ↑C₁
⊢ ∀ ψ ∈ dual (Dual.eval R H) ↑C₂, 0 ≤ ((dualDistrib R G H) (φ ⊗ₜ[R] ψ)) ((TensorProduct.comm R H G) z)

Tactic:
ψ

Goal state after:
case refine_2
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : Module R G
H : Type u_3
inst✝¹ : AddCommGroup H
inst✝ : Module R H
C₁ : PointedCone R G
C₂ : PointedCone R H
z : H ⊗[R] G
hz : ∀ φ ∈ dual (Dual.eval R H) ↑C₂, ∀ ψ ∈ dual (Dual.eval R G) ↑C₁, 0 ≤ ((dualDistrib R H G) (φ ⊗ₜ[R] ψ)) z
φ : Dual R G
hφ : φ ∈ dual (Dual.eval R G) ↑C₁
ψ : Dual R H
⊢ ψ ∈ dual (Dual.eval R H) ↑C₂ → 0 ≤ ((dualDistrib R G H) (φ ⊗ₜ[R] ψ)) ((TensorProduct.comm R H G) z)
Module: Mathlib.Geometry.Convex.Cone.TensorProduct

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : Module R G
H : Type u_3
inst✝¹ : AddCommGroup H
inst✝ : Module R H
C₁ : PointedCone R G
C₂ : PointedCone R H
⊢ map (↑(TensorProduct.comm R G H)) (C₁.maxTensorProduct C₂) = C₂.maxTensorProduct C₁

Tactic:
ext z
  simp only [mem_map, mem_maxTensorProduct]
  refine ⟨?_, fun hz ↦
    ⟨(TensorProduct.comm R H G) z, ?_, (TensorProduct.comm R H G).symm_apply_apply z⟩⟩
  · rintro ⟨w, hw, rfl⟩ ψ hψ φ hφ
    simpa [dualDistrib_apply_comm] using hw φ hφ ψ hψ
  · intro φ hφ ψ hψ
    simpa [dualDistrib_apply_comm] using hz ψ hψ φ hφ

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.TensorProduct

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : LinearOrder R
inst✝⁸ : IsStrictOrderedRing R
G : Type u_2
inst✝⁷ : AddCommGroup G
inst✝⁶ : Module R G
H : Type u_3
inst✝⁵ : AddCommGroup H
inst✝⁴ : Module R H
G' : Type u_4
H' : Type u_5
inst✝³ : AddCommGroup G'
inst✝² : Module R G'
inst✝¹ : AddCommGroup H'
inst✝ : Module R H'
f : G →ₗ[R] G'
g : H →ₗ[R] H'
C₁ : PointedCone R G
C₂ : PointedCone R H
w : G ⊗[R] H
hw : ∀ φ ∈ dual (Dual.eval R G) ↑C₁, ∀ ψ ∈ dual (Dual.eval R H) ↑C₂, 0 ≤ ((dualDistrib R G H) (φ ⊗ₜ[R] ψ)) w
φ : Dual R G'
hφ : φ ∈ dual (Dual.eval R G') ↑(map f C₁)
⊢ ∀ ψ ∈ dual (Dual.eval R H') ↑(map g C₂),
    0 ≤ ((dualDistrib R G' H') (φ ⊗ₜ[R] ψ)) ((↑{ c // 0 ≤ c } (TensorProduct.map f g)) w)

Tactic:
ψ

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : LinearOrder R
inst✝⁸ : IsStrictOrderedRing R
G : Type u_2
inst✝⁷ : AddCommGroup G
inst✝⁶ : Module R G
H : Type u_3
inst✝⁵ : AddCommGroup H
inst✝⁴ : Module R H
G' : Type u_4
H' : Type u_5
inst✝³ : AddCommGroup G'
inst✝² : Module R G'
inst✝¹ : AddCommGroup H'
inst✝ : Module R H'
f : G →ₗ[R] G'
g : H →ₗ[R] H'
C₁ : PointedCone R G
C₂ : PointedCone R H
w : G ⊗[R] H
hw : ∀ φ ∈ dual (Dual.eval R G) ↑C₁, ∀ ψ ∈ dual (Dual.eval R H) ↑C₂, 0 ≤ ((dualDistrib R G H) (φ ⊗ₜ[R] ψ)) w
φ : Dual R G'
hφ : φ ∈ dual (Dual.eval R G') ↑(map f C₁)
ψ : Dual R H'
⊢ ψ ∈ dual (Dual.eval R H') ↑(map g C₂) →
    0 ≤ ((dualDistrib R G' H') (φ ⊗ₜ[R] ψ)) ((↑{ c // 0 ≤ c } (TensorProduct.map f g)) w)
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
⊢ ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
wlog hx : x = 0

Goal state after:
case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E},
    s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
hx : ¬x = 0
⊢ ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x✝ : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
hs : s ∈ 𝓝 x
hx : x = 0
⊢ ∃ b, x ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s
Module: Mathlib.Analysis.Convex.Extrema

Goal state before:
pp: β : Type u_2
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
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsLocalMinOn f (Icc a b) a
h_conv : ConvexOn ℝ (Icc a b) f
⊢ IsMinOn f (Icc a b) a

Tactic:
rintro c hc

Goal state after:
β : Type u_2
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
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsLocalMinOn f (Icc a b) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
⊢ c ∈ {x | (fun x ↦ f a ≤ f x) x}
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
u : Finset E
hu₁ : ↑u ⊆ s
hu₃ : s ⊆ ⋃ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i}
⊢ s ⊆ interior ((convexHull ℝ) (range ⇑b) + (convexHull ℝ) ↑u)

Tactic:
grw [hu₃, ← subset_interior_add_left, Set.iUnion₂_subset_iff, ← subset_convexHull _ (u : Set E)]

Goal state after:
case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
u : Finset E
hu₁ : ↑u ⊆ s
hu₃ : s ⊆ ⋃ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i}
⊢ ∀ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i} ⊆ interior ((convexHull ℝ) (range ⇑b)) + ↑u
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
⊢ ∃ u, s ⊆ interior ((convexHull ℝ) ↑u) ∧ (convexHull ℝ) ↑u ⊆ t

Tactic:
classical
  rcases mem_nhdsSet_iff_exists.1 ht with ⟨U, hU₁, hU₂, hU₃⟩
  rcases compact_open_separated_add_left hs₂ hU₁ hU₂ with ⟨V, hV₁, hV₂⟩
  rcases exists_mem_interior_convexHull_affineBasis hV₁ with ⟨b, hb₁, hb₂⟩
  rcases hs₂.elim_finite_subcover_image (b := s)
      (c := fun x => interior (convexHull ℝ (Set.range b)) + {x})
      (fun _ _ => isOpen_interior.add_right)
      (fun x hx => Set.mem_iUnion₂_of_mem hx <| by simpa using hb₁)
    with ⟨u, hu₁, hu₂, hu₃⟩
  lift u to Finset E using hu₂
  refine ⟨Finset.univ.image b + u, ?_, ?_⟩
  all_goals rw [Finset.coe_add, Finset.coe_image, Finset.coe_univ, Set.image_univ, convexHull_add]
  · grw [hu₃, ← subset_interior_add_left, Set.iUnion₂_subset_iff, ← subset_convexHull _ (u : Set E)]
    intros
    gcongr
    simpa
  · grw [hu₁, hs₁.convexHull_eq, hb₂, hV₂, hU₃]

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

Goal state before:
pp: case refine_2
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
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∑ i ∈ (G n).range, μ.real (⇑(G n) ⁻¹' {i}) = 1

Tactic:
measureReal_def

Goal state after:
case refine_2
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
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∑ i ∈ (G n).range, (μ (⇑(G n) ⁻¹' {i})).toReal = 1