mathlib-initiative/mathlib-tactics huggingface dataset:
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
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x : ι := Classical.arbitrary ι
⊢ (t x).Nonempty

Tactic:
by

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✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
⊢ (x, ↑a) ∉ A

Tactic:
by

Goal state after:
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
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x : ι := Classical.arbitrary ι
y : α
hy : y ∈ t x
ι' : Set ι := {x' | x' ≠ x}
t' : ↑ι' → Finset α := fun x' ↦ (t ↑x').erase y
⊢ Fintype.card ι - 1 = n

Tactic:
by

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

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
l : List P
⊢ List.Sbtw R l ↔ Triplewise (Sbtw R) l ∧ ∀ (a : P), l ≠ [a, a]

Tactic:
by

Goal state after:
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
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x : ι := Classical.arbitrary ι
y : α
hy : y ∈ t x
ι' : Set ι := {x' | x' ≠ x}
t' : ↑ι' → Finset α := fun x' ↦ (t ↑x').erase y
card_ι' : Fintype.card ↑ι' = n
f' : ↑ι' → α
hfinj : Function.Injective f'
hfr : ∀ (x : ↑ι'), f' x ∈ t' x
z₁ z₂ : ι
⊢ ∀ {x : ↑ι'}, y ≠ f' x

Tactic:
by

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

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
p : P
l : List P
⊢ List.Sbtw R (p :: l) ↔ Pairwise (Sbtw R p) l ∧ List.Sbtw R l ∧ l ≠ [p]

Tactic:
by

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✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
v : 𝕜
⊢ v * L (0, 1) = L (0, v)

Tactic:
by

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✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
hv : ∀ (v : 𝕜), v * L (0, 1) = L (0, v)
w : E
h : w ∈ s
⊢ (w, ↑(φ w)) ∈ A

Tactic:
by

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

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'
l : List P
f : P →ᵃ[R] P'
hf : Function.Injective ⇑f
⊢ List.Sbtw R (map (⇑f) l) ↔ List.Sbtw R l

Tactic:
by

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

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'
l : List P
f : P ≃ᵃ[R] P'
⊢ List.Wbtw R (map (⇑f) l) ↔ List.Wbtw R l

Tactic:
by

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

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'
l : List P
f : P ≃ᵃ[R] P'
⊢ List.Sbtw R (map (⇑f) l) ↔ List.Sbtw R l

Tactic:
by

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✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
A : Set (E × 𝕜) := {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}
L : StrongDual 𝕜 (E × 𝕜)
b : ℝ
hLb : re (L (x, ↑a)) < b ∧ ∀ b_1 ∈ {p | p.1 ∈ s ∧ φ p.1 ≤ re p.2}, b < re (L b_1)
u : E →L[𝕜] 𝕜 := L ∘SL inl 𝕜 E 𝕜
c : ℝ := (re (L (0, 1)))⁻¹
z : ↑s
hv : ∀ (v : 𝕜), v * L (0, 1) = L (0, v)
w : E
h : w ∈ s
⊢ re (L (x, 0)) + re (L (0, 1)) * a < re (L (w, 0)) + re (L (0, 1)) * φ w

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
l : List R
h : l.SortedLE
⊢ List.Wbtw R l

Tactic:
by

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

Goal state before:
pp: case refine_2
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x : ι := Classical.arbitrary ι
y : α
hy : y ∈ t x
ι' : Set ι := {x' | x' ≠ x}
t' : ↑ι' → Finset α := fun x' ↦ (t ↑x').erase y
card_ι' : Fintype.card ↑ι' = n
f' : ↑ι' → α
hfinj : Function.Injective f'
hfr : ∀ (x : ↑ι'), f' x ∈ t' x
z : ι
⊢ (fun z ↦ if h : z = x then y else f' ⟨z, h⟩) z ∈ t z

Tactic:
simp only

Goal state after:
case refine_2
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
this✝ : Nonempty ι
this : DecidableEq ι
x : ι := Classical.arbitrary ι
y : α
hy : y ∈ t x
ι' : Set ι := {x' | x' ≠ x}
t' : ↑ι' → Finset α := fun x' ↦ (t ↑x').erase y
card_ι' : Fintype.card ↑ι' = n
f' : ↑ι' → α
hfinj : Function.Injective f'
hfr : ∀ (x : ↑ι'), f' x ∈ t' x
z : ι
⊢ (if h : z = x then y else f' ⟨z, h⟩) ∈ t z
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
l : List P
hl : l ≠ []
h : List.Wbtw R l
this : ∃ l', (∀ a ∈ l', 0 ≤ a) ∧ l'.SortedLE ∧ map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l
⊢ ∃ l', l'.SortedLE ∧ map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l

Tactic:
by

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✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
x : E
a : ℝ
hx : x ∈ s
hax : a < φ x
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
⊢ ∃ l c, s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c ≤ s.domRestrict φ ∧ re (l x) + c = a

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
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 : Pairwise (Wbtw R head) (head2 :: tail) ∧ List.Wbtw R (head2 :: tail)
ht : ¬head2 :: tail = []
hl'' : map (⇑(lineMap ((head2 :: tail).head ht) ((head2 :: tail).getLast ht))) l'' = head2 :: tail
ht2 : tail = []
⊢ (∀ a ∈ [0, 1], 0 ≤ a) ∧
    [0, 1].SortedLE ∧ map ⇑(lineMap head ((head2 :: tail).getLast ht)) [0, 1] = head :: head2 :: tail

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
y : ℝ
⊢ l 1 * y = l y

Tactic:
by

Goal state after:
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
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
ha : ∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
y : ℝ
hy : y ∈ s
⊢ l 1 * y + c' ≤ f y

Tactic:
by

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

Goal state before:
pp: α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑↑s
⊢ #s' ≤ #(s'.biUnion fun a' ↦ t ↑a')

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
⊢ l 1 * x + c' = a

Tactic:
simpa [h1] using hlc'_eq

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 : P
⊢ affineSegment R x y ⊆ ↑line[R, x, y]

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
l : ℝ →L[ℝ] ℝ
c' : ℝ
hlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f
hlc'_eq : re (l x) + c' = a
h1 : ∀ (y : ℝ), l 1 * y = l y
⊢ l 1 * x + c' = a

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
x a : ℝ
hx : x ∈ s
hax : a < f x
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
⊢ ∃ c c', (∀ y ∈ s, c * y + c' ≤ f y) ∧ c * x + c' = a

Tactic:
by

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

Goal state before:
pp: α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
hus : #s = #(s.biUnion t)
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑(↑s)ᶜ
this : DecidableEq ι
⊢ Disjoint s (image (fun z ↦ ↑z) s')

Tactic:
by

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

Goal state before:
pp: α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
hus : #s = #(s.biUnion t)
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑(↑s)ᶜ
this : DecidableEq ι
disj : Disjoint s (image (fun z ↦ ↑z) s')
⊢ #s' = #(s ∪ image (fun z ↦ ↑z) s') - #s

Tactic:
by

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'
f : P →ᵃ[R] P'
x y : P
⊢ ⇑f '' affineSegment R x y = affineSegment R (f x) (f y)

Tactic:
by

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
v : V
⊢ v +ᵥ z ∈ affineSegment R (v +ᵥ x) (v +ᵥ y) ↔ z ∈ affineSegment R x y

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
x : ℝ
hxs : x ∈ s
⊢ f x - 1 < f x

Tactic:
by

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 : V
p : P
⊢ z +ᵥ p ∈ affineSegment R (x +ᵥ p) (y +ᵥ p) ↔ z ∈ affineSegment R x y

Tactic:
by

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
⊢ p -ᵥ z ∈ affineSegment R (p -ᵥ x) (p -ᵥ y) ↔ z ∈ affineSegment R x y

Tactic:
by

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

Goal state before:
pp: s : Set ℝ
f : ℝ → ℝ
hsc : IsClosed s
hfc : LowerSemicontinuousOn f s
hf : ConvexOn ℝ s f
⊢ ∃ c c', ∀ x ∈ s, c * x + c' ≤ f x

Tactic:
by

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
⊢ z -ᵥ p ∈ affineSegment R (x -ᵥ p) (y -ᵥ p) ↔ z ∈ affineSegment R x y

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : PartialOrder R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : IsOrderedRing R
x y : V
⊢ affineSegment R x y = segment R x y

Tactic:
by

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 : AddTorsor V P
inst✝ : IsOrderedRing R
x y : P
⊢ affineSegment R x y = affineSegment R y x

Tactic:
by

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 : AddTorsor V P
inst✝ : IsOrderedRing R
x : P
⊢ affineSegment R x x = {x}

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : PartialOrder R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : IsOrderedRing R
x y z : V
⊢ y ∈ segment R x z ↔ Wbtw R x y z

Tactic:
by

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 : AddTorsor V P
inst✝ : IsOrderedRing R
x y z : P
⊢ Wbtw R x y z ↔ Wbtw R z y x

Tactic:
by

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 : AddTorsor V P
inst✝ : IsOrderedRing R
x y z : P
⊢ Sbtw R x y z ↔ Sbtw R z y x

Tactic:
by

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
s : AffineSubspace R P
x y z : P
hxyz : Wbtw R x y z
hx : x ∈ s
hz : z ∈ s
⊢ y ∈ s

Tactic:
by

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
h : Wbtw R x y z
f : P →ᵃ[R] P'
⊢ Wbtw R (f x) (f y) (f z)

Tactic:
by

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'
hf : Injective ⇑f
⊢ Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z

Tactic:
by

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

Goal state before:
pp: α : Type v
inst✝ : DecidableEq α
ι : Type u
t : ι → Finset α
s : Finset ι
hus : #s = #(s.biUnion t)
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
s' : Finset ↑(↑s)ᶜ
⊢ #s' ≤ #(s'.biUnion fun x' ↦ t ↑x' \ s.biUnion t)

Tactic:
by

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'
hf : Injective ⇑f
⊢ Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z

Tactic:
by

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'
s : AffineSubspace R P
hf : InjOn ⇑f ↑s
hx : x ∈ s
hy : y ∈ s
hz : z ∈ s
⊢ Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z

Tactic:
by

Goal state after:
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
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
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
hs : s.Nonempty
hns : s ≠ univ
hus : #s = #(s.biUnion t)
this : DecidableEq ι
⊢ Fintype.card ↥s ≤ n

Tactic:
by

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'
s : AffineSubspace R P
hf : InjOn ⇑f ↑s
hx : x ∈ s
hy : y ∈ s
hz : z ∈ s
⊢ Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z

Tactic:
by

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'
⊢ Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z

Tactic:
by

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:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
x y z : V
⊢ Wbtw R (-x) (-y) (-z) ↔ Wbtw R x y z

Tactic:
by

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
v : V
⊢ Sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Sbtw R x y z

Tactic:
by

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 : V
p : P
⊢ Sbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ Sbtw R x y z

Tactic:
by

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✝⁸ : RCLike 𝕜
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : Module 𝕜 E
inst✝³ : IsScalarTower ℝ 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
hsc : IsClosed s
hφc : LowerSemicontinuousOn φ s
hφcv : ConvexOn ℝ s φ
⊢ sSup {f | f ≤ s.domRestrict φ ∧ ∃ l c, f = s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c} = s.domRestrict φ

Tactic:
by

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 (p -ᵥ x) (p -ᵥ y) (p -ᵥ z) ↔ Sbtw R x y z

Tactic:
by

Goal state after: