l3lab/ntp-mathlib huggingface dataset:
theorem eval_section_injective_of_eventually_injective {j}
    (Finj : ∀ (i) (f : i ⟶ j), (F.map f).Injective) (i) (f : i ⟶ j) :
    (fun s : F.sections => s.val j).Injective := by
  refine' fun s₀ s₁ h => Subtype.ext <| funext fun k => _
  obtain ⟨m, mi, mk, _⟩ := IsCofilteredOrEmpty.cone_objs i k
  dsimp at h
  

Proof state:
case intro.intro.intro
J : Type u
inst✝¹ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j✝ k✝ : J
s : Set (F.obj i✝)
inst✝ : IsCofilteredOrEmpty J
j : J
Finj : ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)
i : J
f : i ⟶ j
s₀ s₁ : ↑(sections F)
h : ↑s₀ j = ↑s₁ j
k m : J
mi : m ⟶ i
mk : m ⟶ k
h✝ : True
⊢ ↑s₀ k = ↑s₁ k

Next tactic:
rw [← s₀.prop (mi ≫ f), ← s₁.prop (mi ≫ f)] at h
theorem eval_section_injective_of_eventually_injective {j}
    (Finj : ∀ (i) (f : i ⟶ j), (F.map f).Injective) (i) (f : i ⟶ j) :
    (fun s : F.sections => s.val j).Injective := by
  refine' fun s₀ s₁ h => Subtype.ext <| funext fun k => _
  obtain ⟨m, mi, mk, _⟩ := IsCofilteredOrEmpty.cone_objs i k
  dsimp at h
  rw [← s₀.prop (mi ≫ f), ← s₁.prop (mi ≫ f)] at h
  

Proof state:
case intro.intro.intro
J : Type u
inst✝¹ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j✝ k✝ : J
s : Set (F.obj i✝)
inst✝ : IsCofilteredOrEmpty J
j : J
Finj : ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)
i : J
f : i ⟶ j
s₀ s₁ : ↑(sections F)
k m : J
mi : m ⟶ i
h : F.map (mi ≫ f) (↑s₀ m) = F.map (mi ≫ f) (↑s₁ m)
mk : m ⟶ k
h✝ : True
⊢ ↑s₀ k = ↑s₁ k

Next tactic:
rw [← s₀.prop mk, ← s₁.prop mk]
theorem eval_section_injective_of_eventually_injective {j}
    (Finj : ∀ (i) (f : i ⟶ j), (F.map f).Injective) (i) (f : i ⟶ j) :
    (fun s : F.sections => s.val j).Injective := by
  refine' fun s₀ s₁ h => Subtype.ext <| funext fun k => _
  obtain ⟨m, mi, mk, _⟩ := IsCofilteredOrEmpty.cone_objs i k
  dsimp at h
  rw [← s₀.prop (mi ≫ f), ← s₁.prop (mi ≫ f)] at h
  rw [← s₀.prop mk, ← s₁.prop mk]
  

Proof state:
case intro.intro.intro
J : Type u
inst✝¹ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j✝ k✝ : J
s : Set (F.obj i✝)
inst✝ : IsCofilteredOrEmpty J
j : J
Finj : ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)
i : J
f : i ⟶ j
s₀ s₁ : ↑(sections F)
k m : J
mi : m ⟶ i
h : F.map (mi ≫ f) (↑s₀ m) = F.map (mi ≫ f) (↑s₁ m)
mk : m ⟶ k
h✝ : True
⊢ F.map mk (↑s₀ m) = F.map mk (↑s₁ m)

Next tactic:
refine' congr_arg _ (Finj m (mi ≫ f) h)
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  

Proof state:
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
⊢ ∃ a, (fun s => ↑s i) a = x

Next tactic:
let s : Set (F.obj i) := {x}
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  

Proof state:
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
⊢ ∃ a, (fun s => ↑s i) a = x

Next tactic:
haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  

Proof state:
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
⊢ ∃ a, (fun s => ↑s i) a = x

Next tactic:
obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
  

Proof state:
case intro
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
sec : (j : J) → (toPreimages F s).obj j
h : sec ∈ sections (toPreimages F s)
⊢ ∃ a, (fun s => ↑s i) a = x

Next tactic:
refine' ⟨⟨fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk⟩, _⟩
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
  refine' ⟨⟨fun j => (sec j).val, fun jk => by 

Proof state:
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
sec : (j : J) → (toPreimages F s).obj j
h : sec ∈ sections (toPreimages F s)
j✝ j'✝ : J
jk : j✝ ⟶ j'✝
⊢ F.map jk ((fun j => ↑(sec j)) j✝) = (fun j => ↑(sec j)) j'✝

Next tactic:
simpa [Subtype.ext_iff] using h jk
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
  refine' ⟨⟨fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk⟩, _⟩
  · 

Proof state:
case intro
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
sec : (j : J) → (toPreimages F s).obj j
h : sec ∈ sections (toPreimages F s)
⊢ (fun s => ↑s i)
      { val := fun j => ↑(sec j),
        property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =
    x

Next tactic:
have := (sec i).prop
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
  refine' ⟨⟨fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk⟩, _⟩
  · have := (sec i).prop
    

Proof state:
case intro
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this✝ : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
sec : (j : J) → (toPreimages F s).obj j
h : sec ∈ sections (toPreimages F s)
this : ↑(sec i) ∈ ⋂ f, F.map f ⁻¹' s
⊢ (fun s => ↑s i)
      { val := fun j => ↑(sec j),
        property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =
    x

Next tactic:
simp only [mem_iInter, mem_preimage, mem_singleton_iff] at this
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
  refine' ⟨⟨fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk⟩, _⟩
  · have := (sec i).prop
    simp only [mem_iInter, mem_preimage, mem_singleton_iff] at this
    

Proof state:
case intro
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this✝ : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
sec : (j : J) → (toPreimages F s).obj j
h : sec ∈ sections (toPreimages F s)
this : ∀ (i_1 : i ⟶ i), F.map i_1 ↑(sec i) = x
⊢ (fun s => ↑s i)
      { val := fun j => ↑(sec j),
        property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =
    x

Next tactic:
have := this (𝟙 i)
theorem eval_section_surjective_of_surjective (i : J) :
    (fun s : F.sections => s.val i).Surjective := fun x => by
  let s : Set (F.obj i) := {x}
  haveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)
  obtain ⟨sec, h⟩ := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)
  refine' ⟨⟨fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk⟩, _⟩
  · have := (sec i).prop
    simp only [mem_iInter, mem_preimage, mem_singleton_iff] at this
    have := this (𝟙 i)
    

Proof state:
case intro
J : Type u
inst✝³ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s✝ : Set (F.obj i✝)
inst✝² : IsCofilteredOrEmpty J
inst✝¹ : ∀ (j : J), Nonempty (F.obj j)
inst✝ : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
i : J
x : F.obj i
s : Set (F.obj i) := {x}
this✝¹ : ∀ (j : J), Nonempty ((toPreimages F s).obj j)
sec : (j : J) → (toPreimages F s).obj j
h : sec ∈ sections (toPreimages F s)
this✝ : ∀ (i_1 : i ⟶ i), F.map i_1 ↑(sec i) = x
this : F.map (𝟙 i) ↑(sec i) = x
⊢ (fun s => ↑s i)
      { val := fun j => ↑(sec j),
        property := (_ : ∀ {j j' : J} (jk : j ⟶ j'), F.map jk ((fun j => ↑(sec j)) j) = (fun j => ↑(sec j)) j') } =
    x

Next tactic:
rwa [map_id_apply] at this
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i j k : J
s : Set (F.obj i)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
⊢ ∃ j, ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)

Next tactic:
haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i j k : J
s : Set (F.obj i)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
this : (j : J) → Fintype (F.obj j)
⊢ ∃ j, ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)

Next tactic:
haveI : Fintype F.sections := Fintype.ofFinite F.sections
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
  haveI : Fintype F.sections := Fintype.ofFinite F.sections
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i j k : J
s : Set (F.obj i)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
this✝ : (j : J) → Fintype (F.obj j)
this : Fintype ↑(sections F)
⊢ ∃ j, ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)

Next tactic:
have card_le : ∀ j, Fintype.card (F.obj j) ≤ Fintype.card F.sections :=
    fun j => Fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j)
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
  haveI : Fintype F.sections := Fintype.ofFinite F.sections
  have card_le : ∀ j, Fintype.card (F.obj j) ≤ Fintype.card F.sections :=
    fun j => Fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j)
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i j k : J
s : Set (F.obj i)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
this✝ : (j : J) → Fintype (F.obj j)
this : Fintype ↑(sections F)
card_le : ∀ (j : J), Fintype.card (F.obj j) ≤ Fintype.card ↑(sections F)
⊢ ∃ j, ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)

Next tactic:
let fn j := Fintype.card F.sections - Fintype.card (F.obj j)
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
  haveI : Fintype F.sections := Fintype.ofFinite F.sections
  have card_le : ∀ j, Fintype.card (F.obj j) ≤ Fintype.card F.sections :=
    fun j => Fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j)
  let fn j := Fintype.card F.sections - Fintype.card (F.obj j)
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i j k : J
s : Set (F.obj i)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
this✝ : (j : J) → Fintype (F.obj j)
this : Fintype ↑(sections F)
card_le : ∀ (j : J), Fintype.card (F.obj j) ≤ Fintype.card ↑(sections F)
fn : J → ℕ := fun j => Fintype.card ↑(sections F) - Fintype.card (F.obj j)
⊢ ∃ j, ∀ (i : J) (f : i ⟶ j), Function.Injective (F.map f)

Next tactic:
refine' ⟨fn.argmin Nat.lt_wfRel.wf,
    fun i f => ((Fintype.bijective_iff_surjective_and_card _).2
      ⟨Fsur f, le_antisymm _ (Fintype.card_le_of_surjective _ <| Fsur f)⟩).1⟩
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
  haveI : Fintype F.sections := Fintype.ofFinite F.sections
  have card_le : ∀ j, Fintype.card (F.obj j) ≤ Fintype.card F.sections :=
    fun j => Fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j)
  let fn j := Fintype.card F.sections - Fintype.card (F.obj j)
  refine' ⟨fn.argmin Nat.lt_wfRel.wf,
    fun i f => ((Fintype.bijective_iff_surjective_and_card _).2
      ⟨Fsur f, le_antisymm _ (Fintype.card_le_of_surjective _ <| Fsur f)⟩).1⟩
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s : Set (F.obj i✝)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
this✝ : (j : J) → Fintype (F.obj j)
this : Fintype ↑(sections F)
card_le : ∀ (j : J), Fintype.card (F.obj j) ≤ Fintype.card ↑(sections F)
fn : J → ℕ := fun j => Fintype.card ↑(sections F) - Fintype.card (F.obj j)
i : J
f : i ⟶ Function.argmin fn (_ : WellFounded WellFoundedRelation.rel)
⊢ Fintype.card (F.obj i) ≤ Fintype.card (F.obj (Function.argmin fn (_ : WellFounded WellFoundedRelation.rel)))

Next tactic:
rw [← Nat.sub_sub_self (card_le i), tsub_le_iff_tsub_le]
theorem eventually_injective [Nonempty J] [Finite F.sections] :
    ∃ j, ∀ (i) (f : i ⟶ j), (F.map f).Injective := by
  haveI : ∀ j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)
  haveI : Fintype F.sections := Fintype.ofFinite F.sections
  have card_le : ∀ j, Fintype.card (F.obj j) ≤ Fintype.card F.sections :=
    fun j => Fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j)
  let fn j := Fintype.card F.sections - Fintype.card (F.obj j)
  refine' ⟨fn.argmin Nat.lt_wfRel.wf,
    fun i f => ((Fintype.bijective_iff_surjective_and_card _).2
      ⟨Fsur f, le_antisymm _ (Fintype.card_le_of_surjective _ <| Fsur f)⟩).1⟩
  rw [← Nat.sub_sub_self (card_le i), tsub_le_iff_tsub_le]
  

Proof state:
J : Type u
inst✝⁵ : Category.{u_1, u} J
F : J ⥤ Type v
i✝ j k : J
s : Set (F.obj i✝)
inst✝⁴ : IsCofilteredOrEmpty J
inst✝³ : ∀ (j : J), Nonempty (F.obj j)
inst✝² : ∀ (j : J), Finite (F.obj j)
Fsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective (F.map f)
inst✝¹ : Nonempty J
inst✝ : Finite ↑(sections F)
this✝ : (j : J) → Fintype (F.obj j)
this : Fintype ↑(sections F)
card_le : ∀ (j : J), Fintype.card (F.obj j) ≤ Fintype.card ↑(sections F)
fn : J → ℕ := fun j => Fintype.card ↑(sections F) - Fintype.card (F.obj j)
i : J
f : i ⟶ Function.argmin fn (_ : WellFounded WellFoundedRelation.rel)
⊢ Fintype.card ↑(sections F) - Fintype.card (F.obj (Function.argmin fn (_ : WellFounded WellFoundedRelation.rel))) ≤
    Fintype.card ↑(sections F) - Fintype.card (F.obj i)

Next tactic:
apply fn.argmin_le
theorem adjointAux_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪adjointAux A y, x⟫ = ⟪y, A x⟫ := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace G
A : E →L[𝕜] F
x : E
y : F
⊢ inner ((adjointAux A) y) x = inner y (A x)

Next tactic:
rw [adjointAux_apply, toDual_symm_apply, toSesqForm_apply_coe, coe_comp', innerSL_apply_coe,
    Function.comp_apply]
theorem adjointAux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) :
    ⟪x, adjointAux A y⟫ = ⟪A x, y⟫ := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : IsROrC 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace G
A : E →L[𝕜] F
x : E
y : F
⊢ inner x ((adjointAux A) y) = inner (A x) y

Next tactic:
rw [← inner_conj_symm, adjointAux_inner_left, inner_conj_symm]
theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ adjointAux (adjointAux A) = A

Next tactic:
ext v
theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A := by
  ext v
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
v : E
⊢ (adjointAux (adjointAux A)) v = A v

Next tactic:
refine' ext_inner_left 𝕜 fun w => _
theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A := by
  ext v
  refine' ext_inner_left 𝕜 fun w => _
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
v : E
w : F
⊢ inner w ((adjointAux (adjointAux A)) v) = inner w (A v)

Next tactic:
rw [adjointAux_inner_right, adjointAux_inner_left]
@[simp]
theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖adjointAux A‖ = ‖A‖

Next tactic:
refine' le_antisymm _ _
@[simp]
theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by
  refine' le_antisymm _ _
  · 

Proof state:
case refine'_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖adjointAux A‖ ≤ ‖A‖

Next tactic:
refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun x => _
@[simp]
theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by
  refine' le_antisymm _ _
  · refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun x => _
    

Proof state:
case refine'_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : F
⊢ ‖(adjointAux A) x‖ ≤ ‖A‖ * ‖x‖

Next tactic:
rw [adjointAux_apply, LinearIsometryEquiv.norm_map]
@[simp]
theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by
  refine' le_antisymm _ _
  · refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun x => _
    rw [adjointAux_apply, LinearIsometryEquiv.norm_map]
    

Proof state:
case refine'_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : F
⊢ ‖(toSesqForm A) x‖ ≤ ‖A‖ * ‖x‖

Next tactic:
exact toSesqForm_apply_norm_le
@[simp]
theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by
  refine' le_antisymm _ _
  · refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun x => _
    rw [adjointAux_apply, LinearIsometryEquiv.norm_map]
    exact toSesqForm_apply_norm_le
  · 

Proof state:
case refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : IsROrC 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : CompleteSpace E
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖A‖ ≤ ‖adjointAux A‖

Next tactic:
nth_rw 1 [← adjointAux_adjointAux A]