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

Goal state before:
pp: case inl.inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
l : ℝ≥0
x y z : ℝ
hfs : HasConstantSpeedOnWith f (Icc x y) l
hft : HasConstantSpeedOnWith f (Icc y z) l
xy : x ≤ y
zy : z ≤ y
u : ℝ
xu : x ≤ u
uz : u ≤ z
v : ℝ
xv : x ≤ v
vz : v ≤ z
⊢ eVariationOn f (Icc u v) = eVariationOn f (Icc x y ∩ Icc u v)

Tactic:
Icc_inter_Icc,

Goal state after:
case inl.inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
l : ℝ≥0
x y z : ℝ
hfs : HasConstantSpeedOnWith f (Icc x y) l
hft : HasConstantSpeedOnWith f (Icc y z) l
xy : x ≤ y
zy : z ≤ y
u : ℝ
xu : x ≤ u
uz : u ≤ z
v : ℝ
xv : x ≤ v
vz : v ≤ z
⊢ eVariationOn f (Icc u v) = eVariationOn f (Icc (max x u) (min y v))
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: case inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
l : ℝ≥0
x y z : ℝ
hfs : HasConstantSpeedOnWith f (Icc x y) l
hft : HasConstantSpeedOnWith f (Icc y z) l
yx : y ≤ x
u : ℝ
xu : x ≤ u
uz : u ≤ z
v : ℝ
xv : x ≤ v
vz : v ≤ z
⊢ eVariationOn f (Icc x z ∩ Icc u v) = ENNReal.ofReal (↑l * (v - u))

Tactic:
Icc_inter_Icc,

Goal state after:
case inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
l : ℝ≥0
x y z : ℝ
hfs : HasConstantSpeedOnWith f (Icc x y) l
hft : HasConstantSpeedOnWith f (Icc y z) l
yx : y ≤ x
u : ℝ
xu : x ≤ u
uz : u ≤ z
v : ℝ
xv : x ≤ v
vz : v ≤ z
⊢ eVariationOn f (Icc (max x u) (min z v)) = ENNReal.ofReal (↑l * (v - u))
Module: Mathlib.Analysis.ConstantSpeed

Goal state before:
pp: case inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
l : ℝ≥0
x y z : ℝ
hfs : HasConstantSpeedOnWith f (Icc x y) l
hft : HasConstantSpeedOnWith f (Icc y z) l
yx : y ≤ x
u : ℝ
xu : x ≤ u
uz : u ≤ z
v : ℝ
xv : x ≤ v
vz : v ≤ z
⊢ eVariationOn f (Icc u v) = eVariationOn f (Icc y z ∩ Icc u v)

Tactic:
Icc_inter_Icc,

Goal state after:
case inr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
l : ℝ≥0
x y z : ℝ
hfs : HasConstantSpeedOnWith f (Icc x y) l
hft : HasConstantSpeedOnWith f (Icc y z) l
yx : y ≤ x
u : ℝ
xu : x ≤ u
uz : u ≤ z
v : ℝ
xv : x ≤ v
vz : v ≤ z
⊢ eVariationOn f (Icc u v) = eVariationOn f (Icc (max y u) (min z v))
Module: Mathlib.Combinatorics.Hall.Finite

Goal state before:
pp: ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
x : ι
a : α
s' : Finset ↑{x' | x' ≠ x}
this : DecidableEq ι
ha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)
he : s'.Nonempty
h : image (fun z ↦ ↑z) s' = univ
⊢ False

Tactic:
by simpa [← h] using mem_univ x

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✝⁴ : Ring R
inst✝³ : PartialOrder R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y z p : P
⊢ Sbtw R (x -ᵥ p) (y -ᵥ p) (z -ᵥ p) ↔ Sbtw R x y z

Tactic:
rw [Sbtw, Sbtw, wbtw_vsub_const_iff, (vsub_left_injective p).ne_iff,
    (vsub_left_injective p).ne_iff]

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Limits.Konig

Goal state before:
pp: J : Type u
inst✝ : SmallCategory J
F : J ⥤ TopCat
A B : FiniteDiagram J
⊢ ∃ z,
    (fun G ↦ partialSections F G.snd) A ⊇ (fun G ↦ partialSections F G.snd) z ∧
      (fun G ↦ partialSections F G.snd) B ⊇ (fun G ↦ partialSections F G.snd) z

Tactic:
let ιA : FiniteDiagramArrow A.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f =>
    ⟨f.1, f.2.1, Finset.mem_union_left _ f.2.2.1, Finset.mem_union_left _ f.2.2.2.1, f.2.2.2.2⟩

Goal state after:
J : Type u
inst✝ : SmallCategory J
F : J ⥤ TopCat
A B : FiniteDiagram J
ιA : FiniteDiagramArrow A.fst → FiniteDiagramArrow (A.fst ⊔ B.fst) :=
  fun f ↦ ⟨f.fst, ⟨f.snd.fst, ⟨⋯, ⟨⋯, f.snd.snd.snd.snd⟩⟩⟩⟩
⊢ ∃ z,
    (fun G ↦ partialSections F G.snd) A ⊇ (fun G ↦ partialSections F G.snd) z ∧
      (fun G ↦ partialSections F G.snd) B ⊇ (fun G ↦ partialSections F G.snd) z
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 y z
h₂ : Wbtw R w x y
h : y ≠ z
⊢ x ≠ z

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

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✝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
ι : Type u_6
p : ι → P
ha : AffineIndependent R p
w w₁ w₂ : ι → R
s : Finset ι
hw : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
i : ι
his : i ∈ s
r : R
hs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1
hr : ∀ i ∈ s, w i = (r • (w₂ - w₁) + w₁) i
⊢ Sbtw R ((Finset.affineCombination R s p) w₁) ((Finset.affineCombination R s p) (r • (w₂ - w₁) + w₁))
    ((Finset.affineCombination R s p) w₂)

Tactic:
rw [← s.weightedVSub_vadd_affineCombination, s.weightedVSub_const_smul,
    ← s.affineCombination_vsub, ← lineMap_apply, sbtw_lineMap_iff, and_iff_left hs.2,
    ← @vsub_ne_zero V, s.affineCombination_vsub]

Goal state after:
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
ι : Type u_6
p : ι → P
ha : AffineIndependent R p
w w₁ w₂ : ι → R
s : Finset ι
hw : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
i : ι
his : i ∈ s
r : R
hs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1
hr : ∀ i ∈ s, w i = (r • (w₂ - w₁) + w₁) i
⊢ (s.weightedVSub p) (w₁ - w₂) ≠ 0
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
h : ∀ (j : J), ∃ i f, (range ⇑(ConcreteCategory.hom (F.map f))).Finite
j i✝ : J
hi : i✝ ⟶ j
hf : (range ⇑(ConcreteCategory.hom (F.map hi))).Finite
m : Finset (F.obj j)
hmin : ∀ x ∈ {s | ∃ i f, ↑s = range ⇑(ConcreteCategory.hom (F.map f))}, ¬x ⊂ m
i : J
f : i ⟶ j
hm : ↑m = range ⇑(ConcreteCategory.hom (F.map f))
⊢ ∃ i f, ∀ ⦃k : J⦄ (g : k ⟶ j), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map g))

Tactic:
refine ⟨i, f, fun k g =>
    (F.ranges_directed j).directedOn_range.is_bot_of_is_min ⟨⟨i, f⟩, rfl⟩ ?_ _ ⟨⟨k, g⟩, rfl⟩⟩

Goal state after:
J : Type u
inst✝¹ : Category.{v_1, u} J
F : J ⥤ Type v
inst✝ : IsCofilteredOrEmpty J
h : ∀ (j : J), ∃ i f, (range ⇑(ConcreteCategory.hom (F.map f))).Finite
j i✝ : J
hi : i✝ ⟶ j
hf : (range ⇑(ConcreteCategory.hom (F.map hi))).Finite
m : Finset (F.obj j)
hmin : ∀ x ∈ {s | ∃ i f, ↑s = range ⇑(ConcreteCategory.hom (F.map f))}, ¬x ⊂ m
i : J
f : i ⟶ j
hm : ↑m = range ⇑(ConcreteCategory.hom (F.map f))
k : J
g : k ⟶ j
⊢ ∀ a ∈ range fun f ↦ range ⇑(ConcreteCategory.hom (F.map f.snd)),
    a ⊆ (fun f ↦ range ⇑(ConcreteCategory.hom (F.map f.snd))) ⟨i, f⟩ →
      (fun f ↦ range ⇑(ConcreteCategory.hom (F.map f.snd))) ⟨i, f⟩ ⊆ a
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 : AffineSpace V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
s : Simplex R P 1
p : P
⊢ p ∈ s.interior ↔ Sbtw R (s.points 0) p (s.points 1)

Tactic:
by
  rw [interior_eq_image_Ioo, sbtw_iff_mem_image_Ioo_and_ne]
  simp [s.independent.injective.ne (by decide : (0 : Fin 2) ≠ 1)]

Goal state after:
Module: Mathlib.Data.Matrix.PEquiv

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : NonAssocSemiring α
f : l ≃. m
g : m ≃. n
i : l
j : n
⊢ (f.trans g).toMatrix i j = (f.toMatrix * g.toMatrix) i j

Tactic:
rw [toMatrix_mul_apply]

Goal state after:
l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : NonAssocSemiring α
f : l ≃. m
g : m ≃. n
i : l
j : n
⊢ (f.trans g).toMatrix i j = Option.casesOn (f i) 0 fun fi ↦ g.toMatrix fi j
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)
⊢ Sbtw R (t.points i₁) p p₁

Tactic:
have h₁i := h₁.mem_image_Ioo

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 : p₁ ∈ ⇑(lineMap (t.points i₂) (t.points i₃)) '' Set.Ioo 0 1
⊢ Sbtw R (t.points i₁) p p₁
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: 𝕜 : Type u_1
E : Type u
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y ∈ t, f y = 1
g : E → 𝕜
gcombo : ∑ e ∈ t, g e • e = 0
gsum : ∑ e ∈ t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun z ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : ∑ e ∈ t.erase i₀, k e = 1
⊢ ∑ y ∈ t.erase ↑⟨i₀, hi₀⟩, k y = 1

Tactic:
convert! ksum

Goal state after:
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
h✝ : Nonempty n
d : ℕ
ih :
  ∀ m < d,
    ∀ (M : Matrix n n R) (s : R),
      0 ≤ s →
        (∃ M' ∈ doublyStochastic R n, M = s • M') →
          #{i | M i.1 i.2 ≠ 0} = m → ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
hd : #{i | M i.1 i.2 ≠ 0} = d
hs'✝ : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
hs' : 0 ≤ s'
this : ∃ M' ∈ doublyStochastic R n, N = s' • M'
⊢ ∀ a ∈ univ, N a.1 a.2 ≠ 0 → M a.1 a.2 ≠ 0

Tactic:
rintro ⟨i', j'⟩ _ hN' hM'
      have hσ' : σ i' ≠ j' := by
        simpa [Equiv.toPEquiv_apply] using hσ i' j' hM'
      exact hN' <| by simp [N, hM', hσ', Equiv.toPEquiv_apply]

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

Goal state before:
pp: case refine_2
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
w x y z : P
h_ne : x ≠ y
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
hx : x -ᵥ w = t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ w)
ht₂ : t₂ ∈ Set.Icc 0 1
hy : (lineMap x z) t₂ = y
h_denom : 1 - t₁ + t₁ * t₂ ≠ 0
⊢ t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ w - (x -ᵥ w)) + t₁ • t₂ • (x -ᵥ w) = t₁ • t₂ • (z -ᵥ w)

Tactic:
vsub_sub_vsub_cancel_right,

Goal state after:
case refine_2
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
w x y z : P
h_ne : x ≠ y
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
hx : x -ᵥ w = t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ w)
ht₂ : t₂ ∈ Set.Icc 0 1
hy : (lineMap x z) t₂ = y
h_denom : 1 - t₁ + t₁ * t₂ ≠ 0
⊢ t₁ • ((t₂ • (z -ᵥ x) +ᵥ x) -ᵥ x) + t₁ • t₂ • (x -ᵥ w) = t₁ • t₂ • (z -ᵥ w)
Module: Mathlib.Analysis.Convex.Independent

Goal state before:
pp: 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
⊢ ConvexIndependent 𝕜 Subtype.val ↔ ∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t

Tactic:
by
  constructor
  · rintro hc t h x ⟨hxs, hxt⟩
    refine hc { x | ↑x ∈ t } ⟨x, hxs⟩ ?_
    rw [Subtype.coe_image_of_subset h]
    exact hxt
  · intro hc t x h
    rw [← Subtype.coe_injective.mem_set_image]
    exact hc (t.image ((↑) : s → E)) (Subtype.coe_image_subset s t) ⟨x.prop, h⟩

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
tfae_4_to_5 :
  (∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f) →
    ∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
tfae_6_iff_5 :
  (∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f|) ↔
    ∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
h : ∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f|
x : E
hx : x ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map |f| (𝓝 x), (fun x1 x2 ↦ x1 ≤ x2) x r
ε : ℝ
hε : ε > 0
hεD : ball x ε ⊆ C ∩ |f| ⁻¹' {x | (fun x ↦ (fun x1 x2 ↦ x1 ≤ x2) x r) x}
⊢ ∃ K, ∃ t ∈ 𝓝[C] x, LipschitzOnWith K f t

Tactic:
simp only [preimage_ofPred_eq, Pi.abs_apply, subset_inter_iff, hC.nhdsWithin_eq hx] at hεD ⊢

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
tfae_4_to_5 :
  (∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f) →
    ∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
tfae_6_iff_5 :
  (∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f|) ↔
    ∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) f
h : ∀ ⦃x₀ : E⦄, x₀ ∈ C → Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f|
x : E
hx : x ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map |f| (𝓝 x), (fun x1 x2 ↦ x1 ≤ x2) x r
ε : ℝ
hε : ε > 0
hεD : ball x ε ⊆ C ∧ ball x ε ⊆ {a | |f a| ≤ r}
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: case pos
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
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : μ t = 0
⊢ ∫ (x : α) in t, f x ∂μ = μ.real t • ⨍ (x : α), f x ∂μ

Tactic:
rw [restrict_eq_zero.2 h₀, integral_zero_measure, measureReal_def, h₀,
      ENNReal.toReal_zero, zero_smul]

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ∞
d : ℕ := finrank ℝ E
r : ℝ≥0
hr : 1 < r
⊢ ENNReal.ofReal ↑(r ^ finrank ℝ E) * μ (interior s) = ↑(r ^ d) * μ (interior s)

Tactic:
ENNReal.ofReal_coe_nnreal

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ∞
d : ℕ := finrank ℝ E
r : ℝ≥0
hr : 1 < r
⊢ ↑(r ^ finrank ℝ E) * μ (interior s) = ↑(r ^ d) * μ (interior s)
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: case inr
f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Ici y) f
hf_cont : ContinuousWithinAt f (Ici y) y
x : ℝ
hx : x ∈ Ici y
hxy : y < x
h : x ∈ interior (Ici y) → ContinuousWithinAt f (interior (Ici y)) x
⊢ ContinuousWithinAt f (Ici y) x

Tactic:
simp only [nonempty_Iio, interior_Ici', mem_Ioi] at h

Goal state after:
case inr
f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Ici y) f
hf_cont : ContinuousWithinAt f (Ici y) y
x : ℝ
hx : x ∈ Ici y
hxy : y < x
h : y < x → ContinuousWithinAt f (Ioi y) x
⊢ ContinuousWithinAt f (Ici y) x
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: case inr
f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Iic y) f
hf_cont : ContinuousWithinAt f (Iic y) y
x : ℝ
hx : x ∈ Iic y
hxy : x < y
h : x ∈ interior (Iic y) → ContinuousWithinAt f (interior (Iic y)) x
⊢ ContinuousWithinAt f (Iic y) x

Tactic:
simp only [nonempty_Ioi, interior_Iic', mem_Iio] at h

Goal state after:
case inr
f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Iic y) f
hf_cont : ContinuousWithinAt f (Iic y) y
x : ℝ
hx : x ∈ Iic y
hxy : x < y
h : x < y → ContinuousWithinAt f (Iio y) x
⊢ ContinuousWithinAt f (Iic y) x
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: R : Type u_3
X : Type u_5
Y : Type u_6
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
inst✝ : ConvexSpace R Y
f : X → Y
s : Set Y
hf : IsAffineMap R f
hs : IsConvexSet R s
⊢ IsConvexSet R (f ⁻¹' s)

Tactic:
by
  rintro w hw
  simp only [mem_preimage, hf.map_sConvexComb, sConvexComb_map]
  exact hs.iConvexComb_mem fun x hx ↦ hw <| by simpa

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

Goal state before:
pp: case inl.inr.inl
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : x ≤ e
hy : e < y
hc : a • x + b • y ≤ e
⊢ f (a • x + b • y) ≤ a • f x + b • g y

Tactic:
have hc' : a • x + b • e ≤ a • x + b • y := by gcongr

Goal state after:
case inl.inr.inl
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : x ≤ e
hy : e < y
hc : a • x + b • y ≤ e
hc' : a • x + b • e ≤ a • x + b • y
⊢ f (a • x + b • y) ≤ a • f x + b • g y
Module: Mathlib.Analysis.Convex.Quasiconvex

Goal state before:
pp: 𝕜 : Type u_1
β : Type u_3
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → β
inst✝ : LinearOrder β
hf : QuasilinearOn 𝕜 s f
⊢ MonotoneOn f s ∨ AntitoneOn f s

Tactic:
simp_rw [monotoneOn_or_antitoneOn_iff_uIcc, ← segment_eq_uIcc]

Goal state after:
𝕜 : Type u_1
β : Type u_3
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → β
inst✝ : LinearOrder β
hf : QuasilinearOn 𝕜 s f
⊢ ∀ a ∈ s, ∀ b ∈ s, ∀ c ∈ s, c ∈ segment 𝕜 a b → f c ∈ uIcc (f a) (f b)
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: case refine_1
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : s.sum w = 0
h_vsum : ∑ e ∈ s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := {i ∈ s | 0 ≤ w i}
J : Finset ι := {i ∈ s | w i < 0}
p : E := I.centerMass w f
hJI : ∑ j ∈ J, w j + ∑ i ∈ I, w i = 0
hI : 0 < ∑ i ∈ I, w i
hp : J.centerMass w f = p
⊢ p ∈ (convexHull 𝕜) (f '' ↑I)

pp: case refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : s.sum w = 0
h_vsum : ∑ e ∈ s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := {i ∈ s | 0 ≤ w i}
J : Finset ι := {i ∈ s | w i < 0}
p : E := I.centerMass w f
hJI : ∑ j ∈ J, w j + ∑ i ∈ I, w i = 0
hI : 0 < ∑ i ∈ I, w i
hp : J.centerMass w f = p
⊢ p ∈ (convexHull 𝕜) (f '' (↑I)ᶜ)

Tactic:
· exact centerMass_mem_convexHull _ (fun _i hi ↦ (mem_filter.mp hi).2) hI
      (fun _i hi ↦ mem_image_of_mem _ hi)

Goal state after:
case refine_2
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E
s : Finset ι
w : ι → 𝕜
h_wsum : s.sum w = 0
h_vsum : ∑ e ∈ s, w e • f e = 0
nonzero_w_index : ι
h1 : nonzero_w_index ∈ s
h2 : w nonzero_w_index ≠ 0
I : Finset ι := {i ∈ s | 0 ≤ w i}
J : Finset ι := {i ∈ s | w i < 0}
p : E := I.centerMass w f
hJI : ∑ j ∈ J, w j + ∑ i ∈ I, w i = 0
hI : 0 < ∑ i ∈ I, w i
hp : J.centerMass w f = p
⊢ p ∈ (convexHull 𝕜) (f '' (↑I)ᶜ)
Module: Mathlib.Geometry.Convex.ConvexSpace.Module

Goal state before:
pp: R : Type u_2
M : Type u_3
I : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : ConvexSpace R M
inst✝ : IsModuleConvexSpace R M
w : StdSimplex R I
f : I → M
⊢ iConvexComb w f = w.weights.sum fun i r ↦ r • f i

Tactic:
by
  simp [iConvexComb, sConvexComb_eq_sum, Finsupp.sum_mapDomain_index, add_smul]

Goal state after:
Module: Mathlib.Analysis.Convex.SimplicialComplex.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Ring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
s : Finset E
hs : s ∈ K.faces
⊢ (convexHull 𝕜) ↑s ⊆ K.space

Tactic:
convert! subset_biUnion_of_mem hs
  rfl

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: case refine_3
V : Type u
inst✝¹ : Fintype V
inst✝ : DecidableEq V
Adj : V → V → Prop
hs : Std.Symm Adj
hi : Std.Irrefl Adj
⊢ SimpleGraph.mk' ⟨fun v w ↦ decide (Adj v w), ⋯⟩ = { Adj := Adj, symm := hs, loopless := hi }

Tactic:
ext
      simp

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
⊢ G ≠ ⊤ ↔ ∃ a b, a ≠ b ∧ ¬G.Adj a b

Tactic:
simp [eq_top_iff_forall_ne_adj]

Goal state after: