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

Goal state before:
pp: case pos.refine_1
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' = ∅
hsφ : s.domRestrict φ = 0
this : ∀ (a : ↑s), s.domRestrict φ a = 0 a
⊢ ∀ (i : ℕ), s.domRestrict (⇑re ∘ ⇑((fun x ↦ 0) i)) + const (↑s) ((fun x ↦ 0) i) ≤ s.domRestrict φ

pp: case pos.refine_2
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' = ∅
hsφ : s.domRestrict φ = 0
this : ∀ (a : ↑s), s.domRestrict φ a = 0 a
⊢ ⨆ i, s.domRestrict (⇑re ∘ ⇑((fun x ↦ 0) i)) + const (↑s) ((fun x ↦ 0) i) = s.domRestrict φ

Tactic:
·

Goal state after:
case pos.refine_1
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' = ∅
hsφ : s.domRestrict φ = 0
this : ∀ (a : ↑s), s.domRestrict φ a = 0 a
⊢ ∀ (i : ℕ), s.domRestrict (⇑re ∘ ⇑((fun x ↦ 0) i)) + const (↑s) ((fun x ↦ 0) i) ≤ s.domRestrict φ
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: case pos.refine_2
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' = ∅
hsφ : s.domRestrict φ = 0
this : ∀ (a : ↑s), s.domRestrict φ a = 0 a
⊢ ⨆ i, s.domRestrict (⇑re ∘ ⇑((fun x ↦ 0) i)) + const (↑s) ((fun x ↦ 0) i) = s.domRestrict φ

Tactic:
·

Goal state after:
case pos.refine_2
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' = ∅
hsφ : s.domRestrict φ = 0
this : ∀ (a : ↑s), s.domRestrict φ a = 0 a
⊢ ⨆ i, s.domRestrict (⇑re ∘ ⇑((fun x ↦ 0) i)) + const (↑s) ((fun x ↦ 0) i) = s.domRestrict φ
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case pos
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
x y : P
r : R
hxy : x = y
⊢ Wbtw R x ((lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1

pp: case neg
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
x y : P
r : R
hxy : ¬x = y
⊢ Wbtw R x ((lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1

Tactic:
· rw [hxy, lineMap_same_apply]
    simp

Goal state after:
case neg
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
x y : P
r : R
hxy : ¬x = y
⊢ Wbtw R x ((lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case inl
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : IsEmpty J
⊢ F.sections.Nonempty

pp: case inr
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : Nonempty J
⊢ F.sections.Nonempty

Tactic:
·

Goal state after:
case inl
J : Type u
inst✝³ : Category.{w, u} J
inst✝² : IsCofilteredOrEmpty J
F : J ⥤ Type v
inst✝¹ : ∀ (j : J), Finite (F.obj j)
inst✝ : ∀ (j : J), Nonempty (F.obj j)
J' : Type (max w v u) := AsSmall J
down : J' ⥤ J := AsSmall.down
F' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u w, v}
this✝ : ∀ (i : J'), Nonempty (F'.obj i)
this : ∀ (i : J'), Finite (F'.obj i)
h✝ : IsEmpty J
⊢ F.sections.Nonempty
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: 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' = ∅
hsφ : s.domRestrict φ ≠ 0
⊢ ∃ 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:
·

Goal state after:
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' = ∅
hsφ : s.domRestrict φ ≠ 0
⊢ ∃ 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.Topology.Category.TopCat.Limits.Konig

Goal state before:
pp: case refine_2
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⟩⟩⟩⟩
ιB : FiniteDiagramArrow B.fst → FiniteDiagramArrow (A.fst ⊔ B.fst) :=
  fun f ↦ ⟨f.fst, ⟨f.snd.fst, ⟨⋯, ⟨⋯, f.snd.snd.snd.snd⟩⟩⟩⟩
⊢ (fun G ↦ partialSections F G.snd) B ⊇
    (fun G ↦ partialSections F G.snd) ⟨A.fst ⊔ B.fst, Finset.image ιA A.snd ⊔ Finset.image ιB B.snd⟩

Tactic:
·

Goal state after:
case refine_2
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⟩⟩⟩⟩
ιB : FiniteDiagramArrow B.fst → FiniteDiagramArrow (A.fst ⊔ B.fst) :=
  fun f ↦ ⟨f.fst, ⟨f.snd.fst, ⟨⋯, ⟨⋯, f.snd.snd.snd.snd⟩⟩⟩⟩
⊢ (fun G ↦ partialSections F G.snd) B ⊇
    (fun G ↦ partialSections F G.snd) ⟨A.fst ⊔ B.fst, Finset.image ιA A.snd ⊔ Finset.image ιB B.snd⟩
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case a
J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j k : J
f : i ⟶ j
g : j ⟶ k
h : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))
⊢ F.eventualRange k ⊆ range ⇑(ConcreteCategory.hom (F.map (f ≫ g)))

pp: case a
J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j k : J
f : i ⟶ j
g : j ⟶ k
h : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))
⊢ range ⇑(ConcreteCategory.hom (F.map (f ≫ g))) ⊆ F.eventualRange k

Tactic:
·

Goal state after:
case a
J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j k : J
f : i ⟶ j
g : j ⟶ k
h : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))
⊢ F.eventualRange k ⊆ range ⇑(ConcreteCategory.hom (F.map (f ≫ g)))
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: case a
J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j k : J
f : i ⟶ j
g : j ⟶ k
h : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))
⊢ range ⇑(ConcreteCategory.hom (F.map (f ≫ g))) ⊆ F.eventualRange k

Tactic:
·

Goal state after:
case a
J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j k : J
f : i ⟶ j
g : j ⟶ k
h : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))
⊢ range ⇑(ConcreteCategory.hom (F.map (f ≫ g))) ⊆ F.eventualRange k
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:
subset_antisymm_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
⊢ F.eventualRange j ⊆ range ⇑(ConcreteCategory.hom (F.map f)) ∧
      range ⇑(ConcreteCategory.hom (F.map f)) ⊆ F.eventualRange j ↔
    ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: case e'_3
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ι' : Finset ι
s' : Finset ↥ι'
⊢ #s' = #(image Subtype.val s')

pp: case e'_4
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ι' : Finset ι
s' : Finset ↥ι'
⊢ #(s'.biUnion fun i ↦ t ↑i) = #((image Subtype.val s').biUnion t)

Tactic:
·

Goal state after:
case e'_3
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ι' : Finset ι
s' : Finset ↥ι'
⊢ #s' = #(image Subtype.val s')
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: case e'_4
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ι' : Finset ι
s' : Finset ↥ι'
⊢ #(s'.biUnion fun i ↦ t ↑i) = #((image Subtype.val s').biUnion t)

Tactic:
·

Goal state after:
case e'_4
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ι' : Finset ι
s' : Finset ↥ι'
⊢ #(s'.biUnion fun i ↦ t ↑i) = #((image Subtype.val s').biUnion t)
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: case mp
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
⊢ (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) → ∃ f, Injective f ∧ ∀ (x : ι), f x ∈ t x

pp: case mpr
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
⊢ (∃ f, Injective f ∧ ∀ (x : ι), f x ∈ t x) → ∀ (s : Finset ι), #s ≤ #(s.biUnion t)

Tactic:
·

Goal state after:
case mp
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
⊢ (∀ (s : Finset ι), #s ≤ #(s.biUnion t)) → ∃ f, Injective f ∧ ∀ (x : ι), f x ∈ t x
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: case mp
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 ∈ s.closedInterior → p ∈ affineSegment R (s.points 0) (s.points 1)

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:
·

Goal state after:
case mp
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 ∈ s.closedInterior → p ∈ affineSegment R (s.points 0) (s.points 1)
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: case refine_3
𝕜 : Type u_1
E : Type u_2
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
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
𝓕 : Set ({ x // x ∈ univ } → ℝ) := {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c}
this : sSup {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c} = φ ∘ Subtype.val
f : E → ℝ
x✝ : ∃ a ∈ 𝓕, (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
a : ↑univ → ℝ
h : a ≤ φ ∘ Subtype.val
l : E →L[𝕜] 𝕜
c : ℝ
hlc : a = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c
hb : (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
x : E
⊢ f x ≤ φ x

pp: case refine_4
𝕜 : Type u_1
E : Type u_2
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
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
𝓕 : Set ({ x // x ∈ univ } → ℝ) := {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c}
this : sSup {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c} = φ ∘ Subtype.val
f : E → ℝ
x✝ : ∃ a ∈ 𝓕, (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
a : ↑univ → ℝ
h : a ≤ φ ∘ Subtype.val
l : E →L[𝕜] 𝕜
c : ℝ
hlc : a = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c
hb : (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
⊢ f = ⇑re ∘ ⇑l + const E c

Tactic:
·

Goal state after:
case refine_3
𝕜 : Type u_1
E : Type u_2
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
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
𝓕 : Set ({ x // x ∈ univ } → ℝ) := {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c}
this : sSup {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c} = φ ∘ Subtype.val
f : E → ℝ
x✝ : ∃ a ∈ 𝓕, (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
a : ↑univ → ℝ
h : a ≤ φ ∘ Subtype.val
l : E →L[𝕜] 𝕜
c : ℝ
hlc : a = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c
hb : (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
x : E
⊢ f x ≤ φ x
Module: Mathlib.Combinatorics.Hall.Basic

Goal state before:
pp: case mp.refine_1
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ ι → α

pp: case mp.refine_2
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ Injective ?mp.refine_1

pp: case mp.refine_3
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ ∀ (x : ι), ?mp.refine_1 x ∈ t x

Tactic:
·

Goal state after:
case mp.refine_1
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ ι → α
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: case refine_4
𝕜 : Type u_1
E : Type u_2
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
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
𝓕 : Set ({ x // x ∈ univ } → ℝ) := {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c}
this : sSup {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c} = φ ∘ Subtype.val
f : E → ℝ
x✝ : ∃ a ∈ 𝓕, (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
a : ↑univ → ℝ
h : a ≤ φ ∘ Subtype.val
l : E →L[𝕜] 𝕜
c : ℝ
hlc : a = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c
hb : (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
⊢ f = ⇑re ∘ ⇑l + const E c

Tactic:
·

Goal state after:
case refine_4
𝕜 : Type u_1
E : Type u_2
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
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
𝓕 : Set ({ x // x ∈ univ } → ℝ) := {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c}
this : sSup {f | f ≤ φ ∘ Subtype.val ∧ ∃ l c, f = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c} = φ ∘ Subtype.val
f : E → ℝ
x✝ : ∃ a ∈ 𝓕, (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
a : ↑univ → ℝ
h : a ≤ φ ∘ Subtype.val
l : E →L[𝕜] 𝕜
c : ℝ
hlc : a = (⇑re ∘ ⇑l) ∘ Subtype.val + const (↑univ) c
hb : (fun g ↦ g ∘ ⇑(Equiv.Set.univ E).symm) a = f
⊢ f = ⇑re ∘ ⇑l + const E c
Module: Mathlib.CategoryTheory.CofilteredSystem

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

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

Tactic:
·

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

Goal state before:
pp: case mp.refine_2
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ Injective fun i ↦ ↑(u (Opposite.op {i})) ⟨i, ⋯⟩

pp: case mp.refine_3
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ ∀ (x : ι), ↑(u (Opposite.op {x})) ⟨x, ⋯⟩ ∈ t x

Tactic:
·

Goal state after:
case mp.refine_2
ι : Type u
α : Type v
inst✝ : DecidableEq α
t : ι → Finset α
h : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
this✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')
this : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')
u : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFunctor t).obj j
hu : u ∈ (hallMatchingsFunctor t).sections
⊢ Injective fun i ↦ ↑(u (Opposite.op {i})) ⟨i, ⋯⟩
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:
·

Goal state after:
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