l3lab/ntp-mathlib huggingface dataset:
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
⊢ False

Next tactic:
apply not_le_of_lt (Cardinal.cantor α)
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
⊢ 2 ^ α ≤ α

Next tactic:
let yp : C := ∏ fun _ : md => Y
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ 2 ^ α ≤ α

Next tactic:
refine' _root_.trans _ _
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · 

Proof state:
case refine'_1
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ Cardinal.{u}

Next tactic:
exact #(X ⟶ yp)
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · 

Proof state:
case refine'_2
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ 2 ^ α ≤ #(X ⟶ yp)

Next tactic:
apply le_trans (Cardinal.power_le_power_right z)
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        

Proof state:
case refine'_2
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ #(X ⟶ Y) ^ α ≤ #(X ⟶ yp)

Next tactic:
rw [Cardinal.power_def]
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        

Proof state:
case refine'_2
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ #(md → (X ⟶ Y)) ≤ #(X ⟶ yp)

Next tactic:
apply le_of_eq
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        

Proof state:
case refine'_2.a
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ #(md → (X ⟶ Y)) = #(X ⟶ yp)

Next tactic:
rw [Cardinal.eq]
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        

Proof state:
case refine'_2.a
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ Nonempty ((md → (X ⟶ Y)) ≃ (X ⟶ yp))

Next tactic:
refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · 

Proof state:
case refine'_2.a.refine'_1
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ Function.LeftInverse (fun f k => f ≫ Pi.π (fun x => Y) k) Pi.lift

Next tactic:
intro f
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          

Proof state:
case refine'_2.a.refine'_1
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
f : md → (X ⟶ Y)
⊢ (fun f k => f ≫ Pi.π (fun x => Y) k) (Pi.lift f) = f

Next tactic:
ext k
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          

Proof state:
case refine'_2.a.refine'_1.h
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
f : md → (X ⟶ Y)
k : md
⊢ (fun f k => f ≫ Pi.π (fun x => Y) k) (Pi.lift f) k = f k

Next tactic:
simp
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · 

Proof state:
case refine'_2.a.refine'_2
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ Function.RightInverse (fun f k => f ≫ Pi.π (fun x => Y) k) Pi.lift

Next tactic:
intro f
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          

Proof state:
case refine'_2.a.refine'_2
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
f : X ⟶ yp
⊢ Pi.lift ((fun f k => f ≫ Pi.π (fun x => Y) k) f) = f

Next tactic:
ext ⟨j⟩
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          ext ⟨j⟩
          

Proof state:
case refine'_2.a.refine'_2.h.mk
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
f : X ⟶ yp
j : C
snd✝ : (W : C) × (j ⟶ W)
⊢ Pi.lift ((fun f k => f ≫ Pi.π (fun x => Y) k) f) ≫ Pi.π (fun a => Y) { fst := j, snd := snd✝ } =
    f ≫ Pi.π (fun a => Y) { fst := j, snd := snd✝ }

Next tactic:
simp
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          ext ⟨j⟩
          simp
      · 

Proof state:
case refine'_3
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ #(X ⟶ yp) ≤ α

Next tactic:
apply Cardinal.mk_le_of_injective _
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          ext ⟨j⟩
          simp
      · apply Cardinal.mk_le_of_injective _
        · 

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ (X ⟶ yp) → md

Next tactic:
intro f
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          ext ⟨j⟩
          simp
      · apply Cardinal.mk_le_of_injective _
        · intro f
          

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
f : X ⟶ yp
⊢ md

Next tactic:
exact ⟨_, _, f⟩
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          ext ⟨j⟩
          simp
      · apply Cardinal.mk_le_of_injective _
        · intro f
          exact ⟨_, _, f⟩
        · 

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
⊢ Function.Injective fun f => { fst := X, snd := { fst := yp, snd := f } }

Next tactic:
rintro f g k
/-- A small category with products is a thin category.

in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
  ⟨fun r s => by
    classical
      by_contra r_ne_s
      have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
        rw [Cardinal.two_le_iff]
        exact ⟨_, _, r_ne_s⟩
      let md := ΣZ W : C, Z ⟶ W
      let α := #md
      apply not_le_of_lt (Cardinal.cantor α)
      let yp : C := ∏ fun _ : md => Y
      refine' _root_.trans _ _
      · exact #(X ⟶ yp)
      · apply le_trans (Cardinal.power_le_power_right z)
        rw [Cardinal.power_def]
        apply le_of_eq
        rw [Cardinal.eq]
        refine' ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, _, _⟩⟩
        · intro f
          ext k
          simp
        · intro f
          ext ⟨j⟩
          simp
      · apply Cardinal.mk_le_of_injective _
        · intro f
          exact ⟨_, _, f⟩
        · rintro f g k
          

Proof state:
C : Type u
inst✝¹ : SmallCategory C
inst✝ : HasProducts C
X Y : C
r s : X ⟶ Y
r_ne_s : ¬r = s
z : 2 ≤ #(X ⟶ Y)
md : Type u := (Z : C) × (W : C) × (Z ⟶ W)
α : Cardinal.{u} := #md
yp : C := ∏ fun x => Y
f g : X ⟶ yp
k :
  (fun f => { fst := X, snd := { fst := yp, snd := f } }) f = (fun f => { fst := X, snd := { fst := yp, snd := f } }) g
⊢ f = g

Next tactic:
cases k