mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝¹¹ : Ring R
inst✝¹⁰ : PartialOrder R
inst✝⁹ : AddCommGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddTorsor V P
inst✝⁶ : ZeroLEOneClass R
R' : Type u_6
inst✝⁵ : Ring R'
inst✝⁴ : PartialOrder R'
inst✝³ : Module R' V
inst✝² : Module R' R
inst✝¹ : IsScalarTower R' R V
inst✝ : SMulPosMono R' R
x y : P
a : R'
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
⊢ (lineMap x y) (a • 1) = (lineMap x y) a

Tactic:
simp [lineMap_apply]

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

Goal state before:
pp: case neg
α : Type v
inst✝¹ : DecidableEq α
ι : Type u
t : ι → Finset α
inst✝ : Finite ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
val✝ : Fintype ι
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {ι : Type u} {t : ι → Finset α} [Finite ι],
      (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) →
        ∀ (val : Fintype ι), Fintype.card ι = m → ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
hn : Fintype.card ι = n + 1
ih' :
  ∀ (ι' : Type u) [inst : Fintype ι'] (t' : ι' → Finset α),
    Fintype.card ι' ≤ n →
      (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
h : ∃ s, s.Nonempty ∧ s ≠ univ ∧ #(s.biUnion t) ≤ #s
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x

Tactic:
rcases h with ⟨s, sne, snu, sle⟩

Goal state after:
case neg
α : Type v
inst✝¹ : DecidableEq α
ι : Type u
t : ι → Finset α
inst✝ : Finite ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
val✝ : Fintype ι
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {ι : Type u} {t : ι → Finset α} [Finite ι],
      (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) →
        ∀ (val : Fintype ι), Fintype.card ι = m → ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
hn : Fintype.card ι = n + 1
ih' :
  ∀ (ι' : Type u) [inst : Fintype ι'] (t' : ι' → Finset α),
    Fintype.card ι' ≤ n →
      (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
s : Finset ι
sne : s.Nonempty
snu : s ≠ univ
sle : #(s.biUnion t) ≤ #s
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
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
⊢ F.eventualRange j = range ⇑(ConcreteCategory.hom (F.map f)) ↔
    ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))

Tactic:
rw [subset_antisymm_iff, eventualRange, and_iff_right (iInter₂_subset _ _), subset_iInter₂_iff]

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
⊢ (∀ (i_1 : J) (j_1 : i_1 ⟶ j), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map j_1))) ↔
    ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : ∑ i, w i = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Icc 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)

Tactic:
Fin.sum_univ_two

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : w 0 + w 1 = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Icc 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case mpr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
p : P
⊢ p ∈ affineSegment R (s.points 0) (s.points 1) → p ∈ s.closedInterior

Tactic:
rintro ⟨r, ⟨h0, h1⟩, rfl⟩
    rw [← Finset.univ.affineCombination_affineCombinationLineMapWeights _ (Finset.mem_univ _)
      (Finset.mem_univ _), affineCombination_mem_closedInterior_iff
        (Finset.sum_affineCombinationLineMapWeights _ (Finset.mem_univ _) (Finset.mem_univ _) _)]
    intro i
    fin_cases i <;> simp [h0, h1]

Goal state after:
Module: Mathlib.Data.PEquiv

Goal state before:
pp: α : Type u
β : Type v
f : α ≃. β
a₁ a₂ : α
b : β
h₁ : b ∈ f a₁
h₂ : b ∈ f a₂
⊢ a₁ = a₂

Tactic:
← mem_iff_mem

Goal state after:
α : Type u
β : Type v
f : α ≃. β
a₁ a₂ : α
b : β
h₁ : a₁ ∈ f.symm b
h₂ : a₂ ∈ f.symm b
⊢ a₁ = a₂
Module: Mathlib.Data.PEquiv

Goal state before:
pp: case pos
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : b ∈ s
ha : a ∈ s
⊢ some b = some a ↔ some a = some b

pp: case neg
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : b ∈ s
ha : ¬a ∈ s
⊢ some b = some a ↔ False

pp: case pos
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : ¬b ∈ s
ha : a ∈ s
⊢ False ↔ some a = some b

pp: case neg
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : ¬b ∈ s
ha : ¬a ∈ s
⊢ False ↔ False

Tactic:
· simp [eq_comm]

Goal state after:
case neg
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : b ∈ s
ha : ¬a ∈ s
⊢ some b = some a ↔ False

case pos
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : ¬b ∈ s
ha : a ∈ s
⊢ False ↔ some a = some b

case neg
α : Type u
β : Type v
γ : Type w
δ : Type x
s✝ : Set α
inst✝¹ : DecidablePred fun x ↦ x ∈ s✝
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
a b : α
hb : ¬b ∈ s
ha : ¬a ∈ s
⊢ False ↔ False
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : ∑ i, w i = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Ioo 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)

Tactic:
Fin.sum_univ_two

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
s : Simplex R P 1
w : Fin (1 + 1) → R
hw : w 0 + w 1 = 1
h01 : ∀ (i : Fin (1 + 1)), w i ∈ Set.Ioo 0 1
⊢ w = Finset.affineCombinationLineMapWeights 0 1 (w 1)
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
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
⊢ w i₁ ∈ Set.Ioo 0 1

Tactic:
have hs : ∀ i : Fin 3, SignType.sign (w i) = SignType.sign (w i₃) := by
    intro i
    rcases h3 i with (rfl | rfl | rfl)
    · exact h₂s
    · exact h₁s
    · rfl

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
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), SignType.sign (w i) = SignType.sign (w i₃)
⊢ w i₁ ∈ Set.Ioo 0 1
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 = ENNReal.ofReal (dist K L)

Tactic:
exact (ENNReal.ofReal_toReal hausdorffEDist_ne_top).symm

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

Goal state before:
pp: case refine_1
V : Type u_1
inst✝² : SeminormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : T2Space V
u : ℕ → ℝ≥0
K : ConvexBody V
h_zero : 0 ∈ K
hu : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖↑(u n) - ↑0‖ < ε
C : ℝ
hC_pos : C > 0
hC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C
ε : ℝ
hε : 0 < ε
n : ℕ
hn : ‖↑(u n) - ↑0‖ < ε / C
y : V
hyK : y ∈ ↑K
h : (1 + ↑(u n)) • y ∈ ⋂ n, (1 + ↑(u n)) • ↑K
⊢ |↑(u n)| * ‖y‖ < ε

Tactic:
lt_div_iff₀' hC_pos,

Goal state after:
case refine_1
V : Type u_1
inst✝² : SeminormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : T2Space V
u : ℕ → ℝ≥0
K : ConvexBody V
h_zero : 0 ∈ K
hu : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖↑(u n) - ↑0‖ < ε
C : ℝ
hC_pos : C > 0
hC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C
ε : ℝ
hε : 0 < ε
n : ℕ
hn : C * ‖↑(u n) - ↑0‖ < ε
y : V
hyK : y ∈ ↑K
h : (1 + ↑(u n)) • y ∈ ⋂ n, (1 + ↑(u n)) • ↑K
⊢ |↑(u n)| * ‖y‖ < ε
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case inr
R : Type u_1
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x y z : R
hxz✝ : x ≤ z
hxz : x < z
hxz' : 0 < z - x
r : R := (y - x) / (z - x)
hy : y = r * (z - x) + x
⊢ Wbtw R x y z ↔ x ≤ y ∧ y ≤ z

Tactic:
simp [hy, wbtw_mul_sub_add_iff, mul_nonneg_iff_of_pos_right hxz', ← le_sub_iff_add_le,
      mul_le_iff_le_one_left hxz', hxz.ne]

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

Tactic:
t.centerMass_eq_of_sum_1 id hw₂

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 : ∑ i ∈ t, w i • id i = x
t' : Finset E := {i ∈ t | w i ≠ 0}
⊢ ∑ x ∈ t, w x • x = x
Module: Mathlib.Analysis.Convex.Cone.InnerDual

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
C : ProperCone ℝ E
f : E →L[ℝ] F
b : F
seq : ℕ → E
hmem : ∀ (x : ℕ), seq x ∈ C
y : F
hinner : ∀ ⦃x : E⦄, x ∈ C → 0 ≤ ⟪f x, y⟫_ℝ
hx : ∀ (x : ℕ), f (seq x) = (⇑f ∘ seq) x
htends : Filter.Tendsto (⇑f ∘ seq) Filter.atTop (nhds b)
⊢ 0 ≤ ⟪b, y⟫_ℝ

Tactic:
have h n : 0 ≤ ⟪f (seq n), y⟫_ℝ := by simpa [real_inner_comm] using hinner (hmem n)

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
C : ProperCone ℝ E
f : E →L[ℝ] F
b : F
seq : ℕ → E
hmem : ∀ (x : ℕ), seq x ∈ C
y : F
hinner : ∀ ⦃x : E⦄, x ∈ C → 0 ≤ ⟪f x, y⟫_ℝ
hx : ∀ (x : ℕ), f (seq x) = (⇑f ∘ seq) x
htends : Filter.Tendsto (⇑f ∘ seq) Filter.atTop (nhds b)
h : ∀ (n : ℕ), 0 ≤ ⟪f (seq n), y⟫_ℝ
⊢ 0 ≤ ⟪b, y⟫_ℝ
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
z : G ⊗[R] H
⊢ z ∈ C₁.maxTensorProduct C₂ ↔
    ∀ φ ∈ dual (Dual.eval R G) ↑C₁, ∀ ψ ∈ dual (Dual.eval R H) ↑C₂, 0 ≤ ((dualDistrib R G H) (φ ⊗ₜ[R] ψ)) z

Tactic:
simp only [maxTensorProduct, minTensorProduct, dual_hull, mem_dual, Set.forall_mem_image2,
    SetLike.mem_coe, mem_dual]

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s✝ t : Set M
y✝ : N
s : Set M
c : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
y : N
hy : y ∈ {y | ∀ ⦃x : M⦄, x ∈ s → 0 ≤ (p x) y}
x : M
hx : x ∈ s
⊢ 0 ≤ (p x) (↑c • y)

Tactic:
map_smul

Goal state after:
R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s✝ t : Set M
y✝ : N
s : Set M
c : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
y : N
hy : y ∈ {y | ∀ ⦃x : M⦄, x ∈ s → 0 ≤ (p x) y}
x : M
hx : x ∈ s
⊢ 0 ≤ ↑c • (p x) y
Module: Mathlib.Geometry.Convex.Cone.Dual

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s✝ t : Set M
y✝ : N
s : Set M
c : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
y : N
hy : y ∈ {y | ∀ ⦃x : M⦄, x ∈ s → 0 ≤ (p x) y}
x : M
hx : x ∈ s
⊢ 0 ≤ (p x) (c • y)

Tactic:
rw [← Nonneg.coe_smul, map_smul]

Goal state after:
R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s✝ t : Set M
y✝ : N
s : Set M
c : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
y : N
hy : y ∈ {y | ∀ ⦃x : M⦄, x ∈ s → 0 ≤ (p x) y}
x : M
hx : x ∈ s
⊢ 0 ≤ ↑c • (p x) y
Module: Mathlib.Analysis.Convex.Independent

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
hs : ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
t : Set E
ht : t ⊆ s
x : E
hxs : x ∈ s
hxt : x ∈ (convexHull 𝕜) t
h : x ∉ t
⊢ False

Tactic:
exact hs _ hxs (convexHull_mono (Set.subset_sdiff_singleton ht h) hxt)

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

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
s : Set P
⊢ intrinsicClosure 𝕜 s = closure[inst✝¹] s ∩ ↑(affineSpan 𝕜 s)

Tactic:
by
  have h : Topology.IsInducing ((↑) : affineSpan 𝕜 s → P) := .subtypeVal
  rw [intrinsicClosure, h.closure_eq_preimage_closure_image, Set.image_preimage_eq_inter_range,
    Set.image_preimage_eq_of_subset ?_, Subtype.range_coe]
  rw [Subtype.range_coe]
  apply subset_affineSpan

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

Goal state before:
pp: f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Iic y) f
hf_cont : ContinuousWithinAt f (Iic y) y
⊢ ContinuousOn f (Iic y)

Tactic:
by
  intro x hx
  rcases eq_or_lt_of_le (α := ℝ) hx with rfl | hxy
  · exact hf_cont
  · have h := hf_cvx.continuousOn_interior x
    simp only [nonempty_Ioi, interior_Iic', mem_Iio] at h
    rw [continuousWithinAt_iff_continuousAt (Iio_mem_nhds hxy)] at h
    exact (h hxy).continuousWithinAt

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

Goal state before:
pp: X : Type u_2
K : Type u_8
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
w : StdSimplex K X
p : X → Prop
inst✝ : DecidablePred p
hp : ∃ a, p a ∧ w.weights a ≠ 0
⊢ ∀ i ∈ (filter p w.weights).support, 0 < (filter p w.weights) i

Tactic:
by simp +contextual [lt_iff_le_and_ne, eq_comm]

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

Goal state before:
pp: X : Type u_2
K : Type u_8
inst✝⁴ : Semifield K
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : IsDomain K
w : StdSimplex K X
s : Set X
hs : ∃ x ∈ s, w.weights x ≠ 0
inst✝ : DecidablePred fun x ↦ x ∈ s
⊢ ∀ i ∈ (filter (fun x ↦ x ∈ s) w.weights).support, 0 < (filter (fun x ↦ x ∈ s) w.weights) i

Tactic:
by simp +contextual [lt_iff_le_and_ne, eq_comm]

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Semiring R
inst✝⁹ : PartialOrder R
inst✝⁸ : IsStrictOrderedRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : ConvexSpace R M
inst✝² : IsModuleConvexSpace R M
inst✝¹ : ConvexSpace R N
inst✝ : IsModuleConvexSpace R N
f : M → N
hf : IsAffineMap R f
w : StdSimplex R M
⊢ (-f) (sConvexComb w) = sConvexComb (StdSimplex.map (-f) w)

Tactic:
by simp [hf.map_sum_weights, Finsupp.sum_mapDomain_index, add_smul]

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
F : Finset (Set E)
h_card : finrank 𝕜 E + 1 ≤ #F
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_inter : ∀ G ⊆ F, #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
⊢ (⋂₀ ↑F).Nonempty

Tactic:
apply helly_theorem_set' h_convex
  intro I hI_ss hI_card
  obtain ⟨J, _, hJ_ss, hJ_card⟩ := exists_subsuperset_card_eq hI_ss hI_card h_card
  have : ⋂₀ (J : Set (Set E)) ⊆ ⋂₀ I := sInter_mono (by simpa [hI_ss])
  apply Set.Nonempty.mono this
  exact h_inter J hJ_ss (by lia)

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Pow

Goal state before:
pp: case inl
hp₀ : 0 ≤ 0
hp₁ : 0 ≤ 1
⊢ ConcaveOn ℝ≥0 univ fun x ↦ x ^ 0

pp: case inr
p : ℝ
hp₀✝ : 0 ≤ p
hp₁ : p ≤ 1
hp₀ : 0 < p
⊢ ConcaveOn ℝ≥0 univ fun x ↦ x ^ p

Tactic:
· simpa only [rpow_zero] using! concaveOn_const (c := 1) convex_univ

Goal state after:
case inr
p : ℝ
hp₀✝ : 0 ≤ p
hp₁ : p ≤ 1
hp₀ : 0 < p
⊢ ConcaveOn ℝ≥0 univ fun x ↦ x ^ p
Module: Mathlib.Analysis.Convex.SpecificFunctions.Pow

Goal state before:
pp: case inl
hp₀ : 0 ≤ 0
hp₁ : 0 ≤ 1
⊢ ConcaveOn ℝ (Ici 0) fun x ↦ x ^ 0

Tactic:
simpa only [rpow_zero] using! concaveOn_const (c := 1) (convex_Ici _)

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✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
v : V
hv : v ∈ s.direction
⊢ s.WOppSide x (v +ᵥ y) ↔ s.WOppSide x y

Tactic:
rw [wOppSide_comm, wOppSide_vadd_left_iff hv, wOppSide_comm]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hsC : s ⊆ C
hmax : Maximal (fun x ↦ x ∈ S) C
hC : Convex 𝕜 C ∧ Disjoint C t
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
c : E
hc : c ∈ Cᶜ
h : ∀ a ∈ C, segment 𝕜 c a ∩ t = ∅
⊢ False

Tactic:
suffices h : Disjoint (convexHull 𝕜 (insert c C)) t by
    rw [hmax.eq_of_subset ⟨convex_convexHull _ _, h⟩ <|
      (subset_insert ..).trans <| subset_convexHull ..] at hc
    exact hc (subset_convexHull _ _ <| mem_insert _ _)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
hst : Disjoint s t
S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}
C : Set E
hsC : s ⊆ C
hmax : Maximal (fun x ↦ x ∈ S) C
hC : Convex 𝕜 C ∧ Disjoint C t
x : E
hx : x ∈ Cᶜ
y : E
hy : y ∈ Cᶜ
z : E
hz : z ∈ segment 𝕜 x y
hzC : z ∈ C
c : E
hc : c ∈ Cᶜ
h : ∀ a ∈ C, segment 𝕜 c a ∩ t = ∅
⊢ Disjoint ((convexHull 𝕜) (insert c C)) t
Module: Mathlib.Analysis.Convex.Strong

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
a b m : ℝ
x y : E
f : E → ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) +
      m / 2 * (a ^ 2 * ‖x‖ ^ 2 + 2 * (a * (b * inner ℝ x y)) + (b * ‖y‖) ^ 2) =
    a * f x + b * f y - m / 2 * a * b * (‖x‖ ^ 2 - 2 * inner ℝ x y + ‖y‖ ^ 2)

Tactic:
mul_pow

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
a b m : ℝ
x y : E
f : E → ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) +
      m / 2 * (a ^ 2 * ‖x‖ ^ 2 + 2 * (a * (b * inner ℝ x y)) + b ^ 2 * ‖y‖ ^ 2) =
    a * f x + b * f y - m / 2 * a * b * (‖x‖ ^ 2 - 2 * inner ℝ x y + ‖y‖ ^ 2)
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: case refine_1
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ (1 - w i) / w i / ((1 - ε) / ε + (1 - w i) / w i + 1) ≤ 1

Tactic:
refine (div_le_one <| by positivity).2 ?_

Goal state after:
case refine_1
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ (1 - w i) / w i ≤ (1 - ε) / ε + (1 - w i) / w i + 1