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

Goal state before:
pp: case neg
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
head : P
l'' : List R
hl''0 : ∀ a ∈ l'', 0 ≤ a
hl''s : l''.SortedLE
head2 : P
tail : List P
hl : head :: head2 :: tail ≠ []
h : ((∀ a' ∈ tail, Wbtw R head head2 a') ∧ Pairwise (Wbtw R head) tail) ∧ List.Wbtw R (head2 :: tail)
ht : ¬head2 :: tail = []
ht2 : ¬tail = []
hl'' : map (⇑(lineMap head2 (tail.getLast ht2))) l'' = head2 :: tail
hw : Wbtw R head head2 (tail.getLast ht2)
⊢ ∃ l', (∀ a ∈ l', 0 ≤ a) ∧ l'.SortedLE ∧ map (⇑(lineMap head (tail.getLast ht2))) l' = head :: head2 :: tail

Tactic:
rcases hw with ⟨r, ⟨hr0, hr1⟩, rfl⟩

Goal state after:
case neg
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
head : P
l'' : List R
hl''0 : ∀ a ∈ l'', 0 ≤ a
hl''s : l''.SortedLE
tail : List P
ht2 : ¬tail = []
r : R
hr0 : 0 ≤ r
hr1 : r ≤ 1
hl : head :: (lineMap head (tail.getLast ht2)) r :: tail ≠ []
h :
  ((∀ a' ∈ tail, Wbtw R head ((lineMap head (tail.getLast ht2)) r) a') ∧ Pairwise (Wbtw R head) tail) ∧
    List.Wbtw R ((lineMap head (tail.getLast ht2)) r :: tail)
ht : ¬(lineMap head (tail.getLast ht2)) r :: tail = []
hl'' :
  map (⇑(lineMap ((lineMap head (tail.getLast ht2)) r) (tail.getLast ht2))) l'' =
    (lineMap head (tail.getLast ht2)) r :: tail
⊢ ∃ l',
    (∀ a ∈ l', 0 ≤ a) ∧
      l'.SortedLE ∧ map (⇑(lineMap head (tail.getLast ht2))) l' = head :: (lineMap head (tail.getLast ht2)) r :: tail
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 : AddTorsor V P
inst✝ : IsOrderedRing R
x : P
⊢ (fun a ↦ x) '' Set.Icc 0 1 = {x}

Tactic:
(Set.nonempty_Icc.mpr zero_le_one).image_const

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
𝓕' : Set (↑s → ℝ)
h𝓕' :
  𝓕'.Countable ∧
    sSup 𝓕' = s.domRestrict φ ∧ ∀ f ∈ 𝓕', f ≤ s.domRestrict φ ∧ ∃ l c, f = s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c
⊢ ∃ l c,
    (∀ (i : ℕ), s.domRestrict (⇑re ∘ ⇑(l i)) + const (↑s) (c i) ≤ s.domRestrict φ) ∧
      ⨆ i, s.domRestrict (⇑re ∘ ⇑(l i)) + const (↑s) (c i) = s.domRestrict φ

Tactic:
by_cases! he : 𝓕'.Nonempty

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
𝓕' : Set (↑s → ℝ)
h𝓕' :
  𝓕'.Countable ∧
    sSup 𝓕' = s.domRestrict φ ∧ ∀ f ∈ 𝓕', f ≤ s.domRestrict φ ∧ ∃ l c, f = s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c
he : 𝓕'.Nonempty
⊢ ∃ l c,
    (∀ (i : ℕ), s.domRestrict (⇑re ∘ ⇑(l i)) + const (↑s) (c i) ≤ s.domRestrict φ) ∧
      ⨆ i, s.domRestrict (⇑re ∘ ⇑(l i)) + const (↑s) (c i) = s.domRestrict φ

case neg
𝕜 : Type u_1
E : Type u_2
s : Set E
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
𝓕' : Set (↑s → ℝ)
h𝓕' :
  𝓕'.Countable ∧
    sSup 𝓕' = s.domRestrict φ ∧ ∀ f ∈ 𝓕', f ≤ s.domRestrict φ ∧ ∃ l c, f = s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c
he : 𝓕' = ∅
⊢ ∃ l c,
    (∀ (i : ℕ), s.domRestrict (⇑re ∘ ⇑(l i)) + const (↑s) (c i) ≤ s.domRestrict φ) ∧
      ⨆ i, s.domRestrict (⇑re ∘ ⇑(l i)) + const (↑s) (c i) = s.domRestrict φ
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 : P
h₂ : Sbtw R w x y
h₁ : Wbtw R w y x
⊢ False

Tactic:
exact h₂.right_ne ((wbtw_swap_right_iff R w).1 ⟨h₁, h₂.wbtw⟩)

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
w x y z : P
h₁ : Wbtw R z x w
h₂ : Sbtw R z y x
⊢ Sbtw R z y w

Tactic:
exact h₁.trans_sbtw_left h₂

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

Goal state before:
pp: ι : Type u
α : Type v
t : ι → Finset α
ι' : Finset ι
g : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := ⋯
⊢ Injective g

Tactic:
intro f f' h

Goal state after:
ι : Type u
α : Type v
t : ι → Finset α
ι' : Finset ι
g : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩
f f' : ↑(hallMatchingsOn t ι')
h : g f = g f'
⊢ f = f'
Module: Mathlib.Data.PEquiv

Goal state before:
pp: α : Type u
β : Type v
f : α ≃. β
x✝ a✝ : α
⊢ (f x✝).bind ⇑f.symm = some a✝ ↔ (ofSet {a | (f a).isSome = true}) x✝ = some a✝

Tactic:
simp only [eq_some_iff f, Option.isSome_iff_exists, bind_eq_some_iff,
    ofSet_eq_some_iff]

Goal state after:
α : Type u
β : Type v
f : α ≃. β
x✝ a✝ : α
⊢ (∃ a, f x✝ = some a ∧ f a✝ = some a) ↔ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, f a = some a_1}
Module: Mathlib.Data.PEquiv

Goal state before:
pp: case mp
α : Type u
β : Type v
f : α ≃. β
x✝ a✝ : α
⊢ (∃ a, f x✝ = some a ∧ f a✝ = some a) → a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, f a = some a_1}

Tactic:
rintro ⟨b, hb₁, hb₂⟩
    exact ⟨PEquiv.inj _ hb₂ hb₁, b, hb₂⟩

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✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
inst✝ : IsOrderedRing R
n : ℕ
s : Simplex R P n
p : P
i j : Fin (n + 1)
h : i ≠ j
⊢ p ∈ (s.face ⋯).closedInterior ↔ Wbtw R (s.points i) p (s.points j)

Tactic:
rw [s.closedInterior_face_eq_affineSegment h, Wbtw]

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
⊢ (if j ∈ (f i).bind ⇑g then 1 else 0) = Option.rec 0 (fun val ↦ if j ∈ g val then 1 else 0) (f i)

Tactic:
cases f i

Goal state after:
case none
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
⊢ (if j ∈ none.bind ⇑g then 1 else 0) = Option.rec 0 (fun val ↦ if j ∈ g val then 1 else 0) none

case some
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
val✝ : m
⊢ (if j ∈ (some val✝).bind ⇑g then 1 else 0) = Option.rec 0 (fun val ↦ if j ∈ g val then 1 else 0) (some val✝)
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: case inr
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 : ∃ M' ∈ doublyStochastic R n, M = s • M'
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')
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M

Tactic:
exists_mem_doublyStochastic_eq_smul_iff hs

Goal state after:
case inr
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')
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
Module: Mathlib.Analysis.Convex.Body

Goal state before:
pp: case hbc
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)| * C < ε
y : V
hyK : y ∈ ↑K
h : (1 + ↑(u n)) • y ∈ ⋂ n, (1 + ↑(u n)) • ↑K
⊢ ‖y‖ ≤ C

Tactic:
exact hC_bdd _ hyK

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

Goal state before:
pp: 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 : Tendsto u atTop (𝓝 0)
⊢ ⋂ n, (1 + ↑(u n)) • ↑K = ↑K

Tactic:
ext x
  refine ⟨fun h => ?_, fun h => ?_⟩
  · obtain ⟨C, hC_pos, hC_bdd⟩ := K.isBounded.exists_pos_norm_le
    rw [← K.isClosed.closure_eq, SeminormedAddCommGroup.mem_closure_iff]
    rw [← NNReal.tendsto_coe, NormedAddCommGroup.tendsto_atTop] at hu
    intro ε hε
    obtain ⟨n, hn⟩ := hu (ε / C) (div_pos hε hC_pos)
    obtain ⟨y, hyK, rfl⟩ := Set.mem_smul_set.mp (Set.mem_iInter.mp h n)
    refine ⟨y, hyK, ?_⟩
    rw [show (1 + u n : ℝ) • y - y = (u n : ℝ) • y by rw [add_smul, one_smul, add_sub_cancel_left],
      norm_smul, Real.norm_eq_abs]
    specialize hn n le_rfl
    rw [lt_div_iff₀' hC_pos, mul_comm, NNReal.coe_zero, sub_zero, Real.norm_eq_abs] at hn
    refine lt_of_le_of_lt ?_ hn
    gcongr; exact hC_bdd _ hyK
  · refine Set.mem_iInter.mpr (fun n => Convex.mem_smul_of_zero_mem K.convex h_zero h ?_)
    exact le_add_of_nonneg_right (by positivity)

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

Goal state before:
pp: case refine_3
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}
⊢ ∀ (i : ↥t'), 0 < (w ∘ Subtype.val) i

Tactic:
exact fun i =>
      (hw₁ _ (Finset.mem_filter.mp i.2).1).lt_of_ne (Finset.mem_filter.mp i.property).2.symm

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

Goal state before:
pp: case inr
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'
hd' : #{i | N i.1 i.2 ≠ 0} < d
w : Equiv.Perm n → R
hw : ∀ (σ : Equiv.Perm n), 0 ≤ w σ
hw' : ∑ σ, w σ • Equiv.Perm.permMatrix R σ = N
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M

Tactic:
refine ⟨w + fun σ' => if σ' = σ then M i (σ i) else 0, ?_⟩

Goal state after:
case inr
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'
hd' : #{i | N i.1 i.2 ≠ 0} < d
w : Equiv.Perm n → R
hw : ∀ (σ : Equiv.Perm n), 0 ≤ w σ
hw' : ∑ σ, w σ • Equiv.Perm.permMatrix R σ = N
⊢ (∀ (σ_1 : Equiv.Perm n), 0 ≤ (w + fun σ' ↦ if σ' = σ then M i (σ i) else 0) σ_1) ∧
    ∑ σ_1, (w + fun σ' ↦ if σ' = σ then M i (σ i) else 0) σ_1 • Equiv.Perm.permMatrix R σ_1 = M
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: case refine_2
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
σ : Equiv.Perm n
x₁ : Matrix n n R
hx₁ : x₁ ∈ ↑(doublyStochastic R n)
x₂ : Matrix n n R
hx₂ : x₂ ∈ ↑(doublyStochastic R n)
hσ : Equiv.Perm.permMatrix R σ ∈ openSegment R x₁ x₂
i j : n
h₁ : Equiv.Perm.permMatrix R σ i j ∈ openSegment R (x₁ i j) (x₂ i j)
h : x₁ i j ≠ x₂ i j
h₂ : openSegment R (x₁ i j) (x₂ i j) ⊆ Set.Ioo 0 1
⊢ False

Tactic:
specialize h₂ h₁

Goal state after:
case refine_2
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
σ : Equiv.Perm n
x₁ : Matrix n n R
hx₁ : x₁ ∈ ↑(doublyStochastic R n)
x₂ : Matrix n n R
hx₂ : x₂ ∈ ↑(doublyStochastic R n)
hσ : Equiv.Perm.permMatrix R σ ∈ openSegment R x₁ x₂
i j : n
h₁ : Equiv.Perm.permMatrix R σ i j ∈ openSegment R (x₁ i j) (x₂ i j)
h : x₁ i j ≠ x₂ i j
h₂ : Equiv.Perm.permMatrix R σ i j ∈ Set.Ioo 0 1
⊢ False
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case inl.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 : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : ty < 0
hz0 : tz < 0
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

pp: case inl.inr.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 : P
v : V
ty : R
hy0 : ty < 0
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = 0 • v +ᵥ x → ∃ r, p = r • v +ᵥ x
⊢ Wbtw R x (ty • v +ᵥ x) (0 • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (0 • v +ᵥ x) x ∨ Wbtw R (0 • v +ᵥ x) x (ty • v +ᵥ x)

pp: case inl.inr.inr
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 : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : ty < 0
hz0 : 0 < tz
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

Tactic:
· rw [wbtw_comm (z := x)]
      rw [← or_assoc]
      exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonpos _ _ hy0.le hz0.le)

Goal state after:
case inl.inr.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 : P
v : V
ty : R
hy0 : ty < 0
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = 0 • v +ᵥ x → ∃ r, p = r • v +ᵥ x
⊢ Wbtw R x (ty • v +ᵥ x) (0 • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (0 • v +ᵥ x) x ∨ Wbtw R (0 • v +ᵥ x) x (ty • v +ᵥ x)

case inl.inr.inr
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 : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : ty < 0
hz0 : 0 < tz
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case inr.inr.inr.inr
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 : P
v : V
ty tz : R
h : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x
hy0 : 0 < ty
hz0 : 0 < tz
⊢ Wbtw R x (ty • v +ᵥ x) (tz • v +ᵥ x) ∨ Wbtw R (ty • v +ᵥ x) (tz • v +ᵥ x) x ∨ Wbtw R (tz • v +ᵥ x) x (ty • v +ᵥ x)

Tactic:
· rw [wbtw_comm (z := x)]
      rw [← or_assoc]
      exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonneg _ _ hy0.le hz0.le)

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

Goal state before:
pp: E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s t : Set E
x : E
hx : gauge t x ≠ 0
⊢ gauge t (gaugeRescale s t x) = gauge s x

Tactic:
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
    smul_eq_mul, div_mul_cancel₀ _ hx]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : Ring 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : Module 𝕜 E
A : Set E
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hx : x ∈ openSegment 𝕜 x₁ x₂
l : StrongDual 𝕜 E
hAB : IsExposed 𝕜 A {x | x ∈ A ∧ ∀ y ∈ A, l y ≤ l x}
hxB : x ∈ {x | x ∈ A ∧ ∀ y ∈ A, l y ≤ l x}
hl : ConvexOn 𝕜 univ ⇑l
hlx₁ : l x₁ ≤ l x
hlx₂ : l x₂ ≤ l x
y : E
hy : y ∈ A
⊢ l y ≤ l x₁

Tactic:
hlx₁.antisymm (hl.le_left_of_right_le (mem_univ _) (mem_univ _) hx hlx₂)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : Ring 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : Module 𝕜 E
A : Set E
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hx : x ∈ openSegment 𝕜 x₁ x₂
l : StrongDual 𝕜 E
hAB : IsExposed 𝕜 A {x | x ∈ A ∧ ∀ y ∈ A, l y ≤ l x}
hxB : x ∈ {x | x ∈ A ∧ ∀ y ∈ A, l y ≤ l x}
hl : ConvexOn 𝕜 univ ⇑l
hlx₁ : l x₁ ≤ l x
hlx₂ : l x₂ ≤ l x
y : E
hy : y ∈ A
⊢ l y ≤ l x
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: 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
⊢ ε * (f x - f y) ≤ ‖x - y‖ * (f z - f x)

Tactic:
have h := hf.2 hy' hz (by positivity) (by positivity) hab

Goal state after:
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
h : f (a • y + b • z) ≤ a • f y + b • f z
⊢ ε * (f x - f y) ≤ ‖x - y‖ * (f z - f x)
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
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
x : E
hx : x ∈ C
this : ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C
δ : ℝ
hδ₀ : 0 < δ
hδ₁ : δ < 1
y : E := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
hy : y ∈ C
hr✝ : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
ε : ℝ
hε : ε > 0
hr : ∀ ⦃y : E⦄, dist y x₀ < ε → f y ≤ r ∧ y ∈ C
z : E
hz : dist z x < ε * δ
⊢ f z ≤ max r (f y)

Tactic:
have hx₀' : δ⁻¹ • (x - y) + y = x₀ := MulAction.injective₀ (sub_ne_zero.2 hδ₁.ne') <| by
      simp [y, smul_sub, smul_smul, hδ₀.ne', div_eq_mul_inv, sub_ne_zero.2 hδ₁.ne', mul_left_comm,
        sub_mul, sub_smul]

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
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
x : E
hx : x ∈ C
this : ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C
δ : ℝ
hδ₀ : 0 < δ
hδ₁ : δ < 1
y : E := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
hy : y ∈ C
hr✝ : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
ε : ℝ
hε : ε > 0
hr : ∀ ⦃y : E⦄, dist y x₀ < ε → f y ≤ r ∧ y ∈ C
z : E
hz : dist z x < ε * δ
hx₀' : δ⁻¹ • (x - y) + y = x₀
⊢ f z ≤ max r (f y)
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✝ : IsProbabilityMeasure μ
hg : ConcaveOn 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
⊢ ∫ (x : α), g (f x) ∂μ ≤ g (∫ (x : α), f x ∂μ)

Tactic:
by
  simpa only [average_eq_integral] using hg.le_map_average hgc hsc hfs hfi hgi

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
H : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Bornology.IsBounded t → μ (frontier t) = 0
B : ℕ → Set E := fun n ↦ ball x (↑n + 1)
y : E
hy : y ∈ frontier s
N : ℕ := ⌊dist y x⌋₊
⊢ y ∈ B N

Tactic:
simp [B, N, Nat.lt_floor_add_one]

Goal state after: